Is Bard level 20 utterly worthless, mechanically?

The bard's capstone ability is extremely underwhelming

I agree with the other answers that, certainly in combat (and given that D&D is a combat heavy game, not that is has to be, but it typically is and that mindset has informed its design), Superior Inspiration is extremely underwhelming. I can also confirm that there are no bard-specific magic items that need you to have 20 levels in bard, nor does being a level 20 single-classed character impact Epic Boons or anything else like that.

Assuming that by level 20, you've mastered the art of knowing how to pace out your uses of bardic inspiration, the likelihood of you being able to short rest before you run out is very high, unless you intentionally switch up your tactics to try to burn through as many of your bardic inspiration die as possible during combat.

For example, you could start intentionally using a use of bardic inspiration every turn to deal a tonne of psychic damage on your weapon attacks via a College of Whispers bard's Psychic Blades, or by making heavy use of Flourish moves as a College of Swords bard (although by that point you have an infinite pool of d6s for that, so it's literally just if you want that d12 instead of the d6), knowing that you will always have another one should another fight start before your next rest.

I agree with your assessment that, mechanically, you're better off multiclassing into another class (assuming your DM allows multiclassing, but I'm yet to meet one who doesn't). An obvious choice to me would be sorcerer, as you pointed out, since it's a Charisma-based caster like bard, and you get the most number of cantrips from that single level dip (a total of 4).

An alternative would be warlock, another Charisma-based caster, but you'd probably want two levels in that so you can have a couple of Eldritch Invocations as well as the 2 cantrips you'd get (although choosing warlock would deny you your extra 6th and 7th level spell slots as warlocks don't have the Spellcasting feature to stack with the bard's Spellcasting feature when multiclassing, which makes this a poorer choice).

Personally (especially when playing as a more "caster" bard like a College of Lore bard), I find that the bard's selection of cantrips are pretty poor, and you don't want to waste Magical Secrets (or Additional Magical Secrets) picking up cantrips, so I usually end up taking Magic Initiate feat (selecting either from the sorcerer or warlock spell list) just to get hold of a couple of decent cantrips.

Therefore, I would recommend taking that one level dip into sorcerer earlier so that you can benefit from those cantrips immediately. Of course, this does delay your progress to getting wish, etc, by one level...

It's useless, if you mainly use Inspiration in combat

This might seem like a strange statement, since Superior Inspiration triggers upon entering combat, but it's important to keep in mind how this changes out-of-combat use. A 20th level bard can expend all the inspiration they want avoiding a fight, or trying to work their way through a social encounter that would invalidate it. They can do this without worrying about being caught with their proverbial pants down, should things go south.

For some playstyles, that's what a bard does. The bard isn't a fighter (though they absolutely can be.) It's their job to help the party navigate any situation optimally; and if fighting becomes the only remaining option, you now have room to do that, too.

The real question is: What causes Initiative?

This is where things get a little complicated. Whether or not Superior Inspiration does anything might depend on how your DM interprets initiative. According to the PHB, initiative is rolled "when combat starts." It leaves it to the DM to adjudicate when this actually happens in-game.

Suppose that your DM decides that players can "start combat" whenever they want. Perhaps when they go hunting. Perhaps they spar with each other. Perhaps they play a game of Tag. It wouldn't be unreasonable to interpret triggering Superior Inspiration off of initiative rolls as just saying "You always have one use of initiative, as long as you have a few minutes to cool off." After all, the alternative reading is that you get one use of Inspiration whenever the DM wants, which is ...less than satisfying.

In that interpretation, Superior Inspiration becomes wildly powerful out of combat. As long as the checks don't come too close together, the majority of important ability checks your party makes could be made with an average 6.5 boost to the roll. The cost is that in combat you don't have as many to toss around, but you always have one.

In other words, it depends

If combat is your main use for Inspiration, and especially if your DM only rolls initiative when your life is at stake, Superior Inspiration is useless. If your DM chooses to interpret this particular trigger as being anything less than completely literal, it does exactly what a capstone ability is supposed to do: make you as good as possible at being a bard.

Powergaming, Metagaming, and the RAW

To briefly address some of the comments this answer has gathered:

This answer is powergaming. However, so is every possible answer to this question, because that is exactly what the question is asking for. Someone who wasn't powergaming would be choosing class features based on reasons other than utility and mechanical strength.

This answer is not metagaming. It is not metagaming to take advantage of knowledge that your character has. A 20th level bard absolutely understands that they seem to always be able to inspire their friends when something bad goes down. Choosing to not connect the dots just because the reason is spelled out mechanically would be shooting yourself in the foot.

The RAW-ness of this answer is for sure questionable, because the extent to which the DM allows short-cutting like this is table-dependent. As a DM, I'd personally see that the cause of the weirdness here is the rule, not the ruling. Other abilities that trigger at the start of initiative only affect combat (see: Battle Master and Samurai.) The bard's capstone is different. It's taking an ability that can be useful in any situation, and tying it to a feature of the game that's heavily mechanical and is triggered at the DM's whim.

If you assume that player characters understand their own abilities (hard to argue against, at level 20!), and expect abilities to make sense in-universe, then a 20th level bard would naturally want to get into small "fights" constantly. As the comments note, that would be tedious. It's the DM's job to keep the game from being tedious, so they can either make this capstone feature useless, skip the boilerplate sparring, or something in between.

Is there any reason at all to become a level 20 Bard?

Your DM does not allow the optional multiclassing rule.

Other than that, it’s a pretty poor capstone feature.

Yes, it is particularly underwhelming.

Suppose your table actually runs what the Dungeon Master's Guide calls The Adventuring Day:

Assuming typical adventuring conditions and average luck, most adventuring parties can handle about six to eight medium or hard encounters in a day.

Our day will consist of two encounters, a short rest, three encounters, a short rest, and two encounters, then long rest.

On fights 1, 3, and 6, you use 5 Bardic Inspiration, since you got them back on each of your short rests (Font of Inspiration, 5th level bard feature). Note, this is probably not easy to pull off, using Bardic Inspiration is a bonus action, which can only happen once per turn; this means each of these encounters was at least 5 rounds long. So on fights 2, 4, 5, and 7, you regain one for free. Under conditions designed to maximally make use of Superior Inspiration, we got 4 extra Bardic Inspiration. Depending on your subclass, those four extra Bardic Inspiration could be more or less useful (see next section for analysis of subclasses in relation to Superior Inspiration).

But let's be real: no one is burning all of their inspirations in the first fight after a rest if they know there are more fights to come before the next rest.

Realistically, a bard is going to space out the uses of Bardic Inspiration between fights, saving some for out-of-combat situations to get that extra boost on a skill check etc. The only time this ability is going to even be useful is when you go through an unusually high number (4+) of fights and roleplay encounters without taking a short rest. Which can happen, but is not the norm.

Is there really any reason to take this ability at level 20? No. Unless you just really want to be sure you've got that d12 to throw at your friend every time you roll initiative, there are many single level dips that would see considerably more mileage than Superior Inspiration. Like Hexblade. Even if you built a totally spell oriented bard with no combat abilities outside of spellcasting, a single level of Hexblade Warlock can make you competitive as a martial fighter:

At 1st level, you acquire the training necessary to effectively arm yourself for battle. You gain proficiency with medium armor, shields, and martial weapons.

The influence of your patron also allows you to mystically channel your will through a particular weapon. Whenever you finish a long rest, you can touch one weapon that you are proficient with and that lacks the two-handed property. When you attack with that weapon, you can use your Charisma modifier, instead of Strength or Dexterity, for the attack and damage rolls. This benefit lasts until you finish a long rest.

Your puny, clumsy bard who dumped dexterity and strength turns into a martial competitor with the power of charisma. Your AC is probably not great, but that's okay, you now have proficiency with medium armor. Combine this with tenser's transformation from your 14th or 18th level Magical Secrets for proficiency with all weapons and armor as well as a plethora of other powerful benefits and you'll never wonder why you didn't take Superior Inspiration instead.

Analysis of Bardic Colleges in relation to Superior Inspiration.

All bards get the standard Bardic Inspiration feature, which is a pool of dice that can be given to allies so that:

the creature can roll the die and add the number rolled to one ability check, attack roll, or saving throw it makes.

My preceding analysis was loosely based on the utility of this particular feature, but I would now like to go into a more in-depth analysis of how each college's unique Bardic Inspiration ability relates to the 20th level feature, Superior Inspiration. In each case, Superior Inspiration would give you one use of each of these abilities at the beginning of a combat where you were out of Bardic Inspiration.

To be clear, the preceding analysis demonstrates that Superior Inspiration is only marginally useful in a completely contrived scenario - this remains true for all of these subclasses. I maintain that you just won’t get a lot of mileage out of it in normal circumstances.

College of Lore

The College of Lore bard has the ability cutting words:

When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature’s roll.

This is a nice ability to have, but it is situational. You could potentially cause one single attack during an encounter to miss, but it is one you are probably going to save for when an attack is a credible threat of unconsciousness or death.

The Lore Bard's 14th level ability, Peerless Skill, similarly can save your life. It allows you to add Bardic Inspiration to an ability check you make. The utility I like for this one is making your counterspell almost a sure thing - counterspell requires an ability check, and throwing a d12 on top of that check gives you a good shot of politely saying "No thank you" to the Big Bad's 9th level power word kill.

College of Swords

For the College of Swords, Superior Inspiration is almost completely useless. The Sword Bard's 14th level ability Master's Flourish divorces their Blade Flourish abilities from the resource pool of their Bardic Inspiration:

Starting at 14th level, whenever you use a Blade Flourish option, you can roll a d6 and use it instead of expending a Bardic Inspiration die.

The only thing a Swords Bard will ever use Superior Inspiration for is throwing vanilla Bardic Inspiration at their friends. Or if they really want to, they can get a +3 to the average damage of a single blade flourish.

Conveniently enough, the Swords Bard is one that would synergize ridiculously well with a single level of Hexblade Warlock.

College of Valor

This one isn't so bad. The Valor Bard's Combat Inspiration lets their allies add Bardic Inspiration rolls to their damage rolls. An extra 6.5 on average damage during a fight is pretty underwhelming, but the secondary use is similar to the Lore Bard's, but more flexible:

Alternatively, when an attack roll is made against the creature, it can use its reaction to roll the Bardic Inspiration die and add the number rolled to its AC against that attack, after seeing the roll but before knowing whether it hits or misses.

This particular ability is very similar to Cutting Words, but you give it to your friends instead of yourself. Not terrible. Could be good to give to your squishiest ally just in case.

College of Whispers

This one is easily the best. The Whisper Bard's psychic blades is several tiers above every other use of Bardic Inspiration, except possibly the Eloquence bard. How does it synergize with Superior Inspiration?

Every combat you get a fireball's worth of psychic damage that is guaranteed to go off as long as you land at least one hit.

Yes. Please.

College of Glamour

The College of Glamour bard's Mantle of Inspiration is another "really great if you happen to need it" ability. As a bonus action you can give all your friends 14 temp hp and if they want to they can use their reaction to move up to their speed without provoking opportunity attacks. Say a fight isn't going your way, but some positioning rearrangement is really all you need - the Glamour Bard's Mantle of Inspiration has you covered. Pretty situational, but not all that bad. Personally, I'd still rather be a Hexblade.

College of Eloquence

I actually like how this one synergizes with Superior Inspiration. The Eloquence Bard's Infectious Inspiration feature esentially doubles the utility of superior ispiration:

When a creature within 60 feet of you adds one of your Bardic Inspiration dice to its ability check, attack roll, or saving throw and the roll succeeds, you can use your reaction to encourage a different creature (other than yourself) that can hear you within 60 feet of you, giving it a Bardic Inspiration die without expending any of your Bardic Inspiration uses.

You get a 2-for-1 deal on your Bardic Inspiration, and its worth noting that the Eloquence bard's inspiration is a sure thing due to their 6th level feature, Unfailing Inspiration:

Your inspiring words are so persuasive that others feel driven to succeed. When a creature adds one of your Bardic Inspiration dice to its ability check, attack roll, or saving throw and the roll fails, the creature can keep the Bardic Inspiration die.

Overall, the Eloquence Bard takes Superior Inspiration and turns it into something you would expect from a name like Superior Inspiration - it doubles its uses and makes it so it cannot be useless.

College of Creation

The College of Creation Bardic Inspiration feature boosts the utility of the vanilla Bardic Inspiration:

• When the creature rolls the Bardic Inspiration die to add it to an ability check, the creature can roll the Bardic Inspiration die again and choose which roll to use, as the mote pops and emits colorful, harmless sparks for a moment.
• Immediately after the creature rolls the Bardic Inspiration die to add it to an attack roll against a target, the mote thunderously shatters. The target and each creature of your choice that you can see within 5 feet of it must succeed on a Constitution saving throw against your spell save DC or take thunder damage equal to the number rolled on the Bardic Inspiration die.
• Immediately after the creature rolls the Bardic Inspiration die and adds it to a saving throw, the mote vanishes with the sound of soft music, causing the creature to gain temporary hit points equal to the number rolled on the Bardic Inspiration die plus your Charisma modifier (minimum of 1 temporary hit point).

This is the only College of Creation feature that relies on your Bardic Inspiration, so using vanilla inspiration is all you're doing with it anyway. Given that these features are only marginal improvements to Bardic Inspriation, I'd say Superior Inspiration is pretty lackluster on a Creation bard.

College of Spirits

This one actually works pretty well. The Spirits bard uses Bardic Inspiration to roll on the Spirit Tales table, and their 14th level feature allows you to roll twice and pick the result:

Mystical Connection

You now have the ability to nudge the spirits of Tales from Beyond toward certain tales. Whenever you roll on the Spirit Tales table, you can roll the die twice and choose which of the two effects to bestow. If you roll the same number on both dice, you can ignore the number and choose any effect on the table.

Some of the available effects on this table are actually quite good, here are a couple of my favorites:

Tale of the Mind-Bender. You evoke an incomprehensible fable from an otherworldly being. The target must succeed on an Intelligence saving throw or take psychic damage equal to three rolls of your Bardic Inspiration die and be stunned until the end of its next turn.

Tale of the Dragon. The target spews fire from the mouth in a 30-foot cone. Each creature in that area must make a Dexterity saving throw, taking fire damage equal to four rolls of your Bardic Inspiration die on a failed save, or half as much damage on a successful one.

Tale of the Beguiler. The target must succeed on a Wisdom saving throw or take psychic damage equal to two rolls of your Bardic Inspiration die, and the target is incapacitated until the end of its next turn.

Some of these features are quite powerful, particularly those that can incapacitate a target for a turn. Having a guaranteed use of this feature every combat is quite useful if you're using your available Bardic Inspirations liberally.