I wanted to go a bit of a different direction with my answer and say that the light hammer does have some special considerations, even if it doesn't technically have a second property to describe them.

Of the comparable items you list, only the hand-axe could be thought of as a tool, and for that it would most likely be used for chopping wood or cutting ropes.

The light hammer is primarily a tool. Carry around a bunch of nails and you can wreak all kinds of havoc. Nail doors shut. Pry a loose floorboard up and hide under it. Destabilize wooden constructions (possibly catapults? or mine-shaft support beams?) Hang a picture to hide a secret on the wall. Pretend you're a doctor and test someone's knee reflexes but actually crush their knee cap... the opportunities are endless!

"Broken", at least as I have typically heard it, generally is used for things that are overpowered so that they crowd out the strategy space and there are no downsides or balancing counters. I would edit your title to just asking whether light hammers are underpowered.

My initial assumption would be that hammers are a strength weapon, and that bludgeoning weapons should not have finesse, but I don't have @Sam Lacrumb's experience in actually using one. I would also assume (without experience) that using a hammer as a finesse weapon to attack vulnerable areas like the collar bone would largely be offset by armor specifically protecting those areas. Thus one thing you might consider is giving the light hammer the finesse property but only when used against natural or light armor, not medium or heavy. This would give it a substantial improvement to its current status without becoming better in every situation.

Another conditional possibility that would add race flavor and verisimilitude considering the physics of wielding one overhand would be to not add finesse, but to bump the damage to a d6 when used against opponents of a smaller size class, or the same size class when mounted (thus dwarves would get a d6 against Small opponents (goblins) on the ground, and against Medium opponents when using them while mounted on mules or ponies). And / or bump their damage to a d6 and their range to 30/90 when used mounted so long as they were thrown in the direction of the mount's movement and the mount had moved for at least half its speed before the attack, to simulate adding the momentum of the mount's movement to the attack.

The light hammer is supposed to suck.

It has the following properties:

• Simple one-handed weapon.
• 1d4 bludgeoning.
• Throwable (range 20/60).

This is suspiciously similar to the properties of an improvised weapon:

An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.

From this I deduce that the light hammer is a construction tool, not a real weapon. It deals improvised weapon damage and has the improvised weapon throwing range. Note that the only NPCs who have "light hammer" on their statblocks are the Pirate Bosun (Saltmarsh) and the Scorchbringer Guard (Ravnica)--folks who do mechanical repairs. Soldiers do not carry this.

(This should not be confused with a warhammer, which is very much a soldier's weapon--it's labeled in the PHB as a "war pick", a martial weapon, 1d8 piercing damage. This illustration at least (which I believe is from 4e) clearly shows a warhammer.)

The only effect of the light hammer appearing in the weapon table is to make simple weapon proficiency apply to it, so that when someone picks up a carpenter's hammer or a tent peg mallet as a weapon, they can say "it's similar to a light hammer" and be proficient.

As requested, this is my experience actually fighting with a war hammer and a pike both on and off horseback in real life.

You need a basic amount of strength to wield the weapon. However, in combat it's mostly an agility weapon. Precision on hits is far more important than brute strength, and the reason the weapons are effective is because you can focus a lot of force in a precise area.

I understand that D&D is not a perfect real world translation. But if you are looking for a reason to make warhammers Finesse weapons, let me assure you that they are. Or at least can be. Its not like the movies at all.

Having said that I have never fought with a Maul or axes they are outside my experience.

Also worth noting, a rapier weighs about the same as a arming sword (d&d long sword). and warhammers are only slightly heavier than those.

Not an imbalancing idea, but a... strange idea

Hammers are about hammering stuff with your strength. Their damage is Bludgeoning for a reason. If you consider other weapons, Finesse is never given to Bludgeoning damage-type weapons, simply because Bludgeoning and Finesse are pretty much on the opposite end of how the character handles the weapon.

There is absolutely no reason, however, that this will be broken. I strongly doubt that even giving it 1d6 damage would make it broken in any way. Bludgeoning is, indeed, overall a better damaging option. Monsters don't have immunity to bludgeoning, and some have vulnerability, while Slashing has its problems against that damned Black Pudding. However, do people stop using Greatswords just because Maul is possibly overall better? I doubt it. My current Paladin uses a Greatsword. And I made that choice knowing the dungeon I was going to play has skeletons and black puddings.

You contradict yourself

As written, I can't see why anyone would choose it.

For characters of dwarven ancestry, for example, it seems more race-appropriate to use light hammers

There you go: that is the reason a character may choose a Light Hammer over a Handaxe or a Javelin: thematic reasons. Not every character has to use the optimal choice of weapon to deal damage. Otherwise, only Barbarians would pick Greataxes, if anyone.

Lots of bad weapons

And let us be fair: there are a bunch of bad weapons around there, including weapons strictly worse than others, and even worse than the Light Hammer. The Club is a Light Hammer without Thrown property. The Sickle is a Club with Slashing damage, which as we just stated, is overall worse. The Mace is a worse, more expensive Quarterstaff. The Morningstar is literally a War Pick, at three times the cost and twice the weight.

Most of these weapons are there to allow thematic choices, not optimal choices.

If you really want to change it

I would go with the 1d6 idea rather than the Finesse idea. I mean, it's not like the Dwarven Rogue is an optimal class-race choice either way, so I'm assuming that if you care about optimization of your weapons, that's not the actual use-case.

Ignoring price, the Player's Handbook has a lot of sub-optimal weapons; they don't need to be fixed

Examples:

The Dagger is better than every single 1d4 weapon (Club, Light Hammer, Sickle)

The Light Hammer is better than the Club and the Sickle

Every single 1d6 weapon (Handaxe, Javelin, Quarterstaff, Spear) is better than the Mace

The Spear is better than the Quarterstaff

The Battleaxe, Longsword, Warhammer, and Rapier are better than the Flail, Morningstar, and War Pick

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A "bad" weapon doesn't need to be fixed

A lot of these sub-optimal weapons are still cool or useful for character concepts; they just aren't perfectly optimized choices, which aren't things everybody needs to be making at most tables. If you're at a table that requires optimal weapon choices, then a lot of weapons are going to be flat out unusable.

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Fixing "bad" weapons takes a lot of work

Fixing these (making each weapon have some reason to use it over the others in the same Simple/Martial grouping) requires significant overhaul and changes to the entire weapons system. I have experience with helping build a system for fifth edition D&D that does exactly this and the system is a lot more complicated than what the PHB uses.

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Do what you want and test things out

At the end of the day, a bump in your damage die isn't going to do a whole lot. You're gaining less than 1 additional damage on average (you aren't guaranteed to hit) which isn't very much, it's actually the least amount of extra damage possible at all.

If you're thinking about adding damage or some property to a weapon, I'll always recommend testing it out and seeing how it goes. In this case, I don't believe anything urgently needs fixing (if it does, then lots of other things do too). I would probably recommend increasing the damage die to make it more similar to a Spear, if I had to recommend some sort of change.

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Damage type is mostly meaningless

From all the books, the only monsters that are immune to one of bludgeoning, piercing, and slashing and not the others are the Black Pudding, Elder Black Pudding, Intelligent Black Pudding, Reduced-Threat Black Pudding, Ochre Jelly, Huge Ochre Jelly, Sentient Ochre Jelly, and Reduced-Threat Oche Jelly (they are all immune to slashing).

From all the books, the only monsters that resist one of bludgeoning, piercing, and slashing and not the others are the Geist and Topi (resist bludgeoning); the Animated Tile Chimera, Awakened Shrub, Xorn, Big Xorn, Flameskull, Thorny, Vegepygmy, and Vegepygmy Chief (resist piercing); and the Warlock of the Fiend (resists slashing).

From all the books, the only monsters that are vulnerable to one of bludgeoning, piercing, and slashing and not the others are the Skeleton, Skeletal Giant Owl, Skeleton Key, Skeletal Alchemist, Warhorse Skeleton, Blazing Skeleton, Minotaur Skeleton, Ogre Skeleton, Skeletal Polar Bear, Skeletal Swarm, Skeletal Owlbear, Skeletal Two-Headed Owlbear, Thunderbeast Skeleton, Giant Shark Skeleton, Skeletal Juggernaut, Giant Skeleton, Ice Mephit, and Stone Cursed (vulnerable to bludgeoning); and the Mordakhesh, Rakshasa, and Zakya Rakshasa (vulnerable to slashing from a magic weapon wielded by a Good creature).

Unless your campaign is absolutely full of Skeletons or Ochre Jellies and Black Puddings, the difference between bludgeoning, piercing, and slashing damage is almost always meaningless. There are 2020 or so monsters, and only about 40 of them are going to make a difference between damage types even exist.

Not all weapons are equally powerful and they don't need to be

The light hammer is not the only weapon that you would never use from a perspective of power.

The sickle is equally bad as the light hammer. Similarly, there is really no reason to specifically use a trident. It is the same as a spear but needs martial weapons proficiency.

When you start to consider that every weapon should have their advantages, you have even more work.

Then again, I don't see why it is absolutely necessary to only have weapons that have a use for the optimizer.

Finesse seems very non-thematic on a hammer

While adding finesse to the light hammer won't have many mechanical consequences, it means that Dex based characters now have a finesse bludgeoning weapon against skeletons. It is true that this probably won't come up that often and you said you don't find this important but it is a fact. It won't break your game.

On the other hand, as stated above, I don't think there is a problem to be fixed. And adding finesse to a hammer seems very non-thematic.