My Current Final Answer to Caster/Martial Disparity

[Ignimortis]

Of course casting spells make you worse. A wizard with few spells is less powerful than a wizard with more spells.
It seems to me that the problem isn't that, but it's simply that the Berserker rage costs too much for what it offers. But that's just a problem with the specific feature.

Regarding wizards getting exhaustion from casting spells: in Ars Magica it works like that. In older D&D editions certain spells have even worse prices, like aging you.
But wizards are still kings in those games, because the spells are just that good.
It's a matter of appropriate prices (and risks) for appropriate rewards.

In part, yes, that's the question about the price being too steep for the effect provided. But the wizard with less spells is worse at only one thing - spellcasting. All of his other capabilities are untouched. The berserker who just used his rage is worse at things that are not necessarily related to combat.

Your distinction between limitations and punishments is completely arbitrary and doesn't really make sense.
Consider this: losing HP (in D&D I presume) doesn't give you a penalty to any roll. So why do you list it as a punishment?
Yes, a character with fewer HP is more likely to die. But the same is true for a wizard with fewer spell slots.

Because HP in general isn't bound to any resource-expenditure mechanics, and is a resource that you don't choose to use - you just get hit or not. Therefore, any mechanic that makes you spend HP to achieve an effect, introduces an active way to spend a resource that wasn't designed to be spent actively, and is in fact more paramount to your survival than any other resource.