What would you change in Heroes of Might and Magic IV?

The problem is that in the third game, there were UPGRADES. A nasty, annoying, and useless feature that doesn't add anything useful to the game.

Well, that's a bit harsh. Upgrades were in all parts of Heroes, except for the 4th, it's a feature of classic HoMM. It's strange to object to one of the typical features of the game, in general.
As for adding something useful - at the very least, it's one of the components of the game's economy. For example, you can build all the creatures in a Castle and admire their growth, while postponing the upgrade for later, investing money, for example, in another Castle, in buying artifacts, in developing magic, finally. Or you can immediately invest money in some upgrade that seems most important to you for quickly clearing the map.
I'm not even talking about the insane amount of a certain resource required to upgrade level 7 creatures - the pursuit of an extra crystal or gem significantly changes the player's actions, up to choosing the path to clear the map, if the coveted pile of resources looms ahead.
If you remember, the exchange rates in the third game are not as pleasant as in the fourth, so the resources spent on the Guild of Magic and upgrades are a very important strategic component.

Because of these upgrades, there were a whole bunch of problems: Necromancers couldn't turn skeletons into skeleton warriors (because the "Necromancy" skill only made regular skeletons, which, without a special slot in the army, simply disappeared).

It should be noted that at least in SoD (and maybe already in AB, I'm not sure), if there is no hero-necromancer in the army and there is a stack of skeleton warriors after the battle, he receives именно upgraded skeletons.
And in general, what kind of strategy is without problems? Without problems - Barbie Fashion Designer.