No, it makes spellcasters more powerful, relatively speaking

It's true spells that last 1 minute are now horrible, but there are lots of other spells. Additionally, some spells acquire new abilities, principally the ability to detect how many creatures are in an area (one assumes the entire world does not run on a global initiative or else all but the longest-lived races would immediately die).

As a Warlock, for example, you may have access to an at-will version of the Sending spell via invocation. In the new game, that spell lets you perfectly identify the number of creatures around any area you can affect with it on any plane of existence (It takes two castings to do so, though-- one to get the sum of the creatures in your and the target's areas and one to get the sum of the creatures in your and a controlled area). Additionally, depending on how 'being in the round' works, you can probably indirectly identify the exact location and size (treating 'medium or smaller' as one size) of each creature in the area with no chance of failure or being detected. Knowing the exact layout and approximate composition of every enemy force you face is quite a potent ability.

Some other spells, like those which enable spurts of quick movement (e.g. teleport), are also made better, albeit less obviously, in this new game. They allow you to waylay enemy formations and cause them absurd travel delays they have practically no way to respond to, while still being able to get out when you need to. For example, you could hide in a bush near the road an enemy scout is walking along (armies can't move, so travelling with more than one creature is only valuable for stopping enemy movement) with your traditional bag o' rats, toss it gently towards the scout, ready a casting of teleport for when an increment of 10 seconds passes, and then leave the morass at any time, whereas the scout has to get far enough away from the rats that his 90 ft/round movement isn't less than 1 foot a year or whatever.

Spellcasters also fare better when traveling alone than many other classes, and this new system requires that players travel alone and never form into parties except for combat that cannot be handled individually nor avoided. This is more true for divine spellcasters because Wisdom checks are used to detect danger and such, but spellcasting generally provides a lot of general utility, and that is what's going to be most useful in this new game. Any time you with any other character, even a familiar, you are less than half as effective as you typically would be. The only way around this is to be part of a group initiative that shares a single turn, but not many player-facing options exist to do that without at least two initiatives and thus half efficiency at best. Simulacrum is the main exception and remains as broken as ever, but you don't get that spell.

Warlocks also have a new interesting strategy in this new game you may wish to try out. They alone refresh spell slots on a short rest, which takes an hour. So long as you are currently involved in a battle near at least 361 total creatures, everyone can be completing a short rest every round, which benefits you more than most characters (Fighters, most importantly Eldritch Knights because no spellcasting = no teleportation, also benefit hugely from such an encounter). A typical spellcaster would have to rest 8ish rounds in such a fight to get their spells back, which gives you plenty of time to defeat them, nova-ing your most powerful spells every round. You need to look out for battles with 8641+ creatures in the area, though: those let everyone long rest every round, which makes coordinated action from primary spellcasters specialized in blasting the only real way to make progress in the fight. Such fights will likely take years anyways, so, as previously suggested, you may benefit more by using your class features to escape via teleportation than by trying to fight it out.

Spells are also required for long-distance communication, creature-less food production, asexual reproduction, and many other concerns created by this new world where creatures want to avoid directly contacting each other. Everyone needs to be alone and everyone needs spells, so I'd argue this rule change makes spellcasters better not worse.

Spell durations assume 6 seconds per round.

First, the Player's Handbook says concerning turns and rounds:

A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn.

One important thing to note here is that turns and rounds are both 6 seconds long. The idea is that all the turns happen concurrently in some sense during the six second round. It seems that your DM is fundamentally misunderstanding this point.

That being said, the spell durations were written assuming that a round is 6 seconds long. Therefore, you are correct to observe that this rule fundamentally destroys the meaning behind a spell duration of 1 minute.

This is definitely a talk to your DM situation. I can tell you that this is fundamentally game-breaking, but I cannot tell you not to play this way - that's between you and your DM. Many spells will be severely nerfed with this rule, and both you and your DM need to understand this fact before moving forward.

Additionally, the fundamental misunderstanding of making turn length add up to round length could have unforeseen consequences that might totally break combat.

For some perspective, read this Q&A concerning changing the length of a round to five seconds. Changing round length by a single second make significant changes to the combat system. What your DM is proposing is erasing the entire system and replacing it with something that has never been tested and using it in conjunction with rules that were written for a completely different system. It's just a terrible idea all around.