They are not human--but they are sapient beings with self-determination and personal identity. Why is it their sapience, self-determination, and personal identity are always trumped by their physiology?
Completely missing the point. I'm not saying every race should be "can do a bit of everything." I'm saying that every race that is capable of being a player character should have sapience, self-determination, and personal identity. Sapience being human-like awareness and intelligence: the ability to understand oneself as distinct from one's environment, to reason about the world around oneself, etc. Every playable species should have this trait in order to be, y'know, playable. Self-determination should, I hope, be self-evident, but just in case, it means the ability to make one's own choices for how one will live, what actions one will take, etc. (Obviously, some situations like dictatorial regimes or slavery or the like can severely degrade a person's ability to exercise their self-determination, but they have said determination nonetheless.) And, finally, a personal identity is individuation: all the personality quirks, verbal tics, tastes, preferences, dislikes, turns of phrase, etc. which make a person distinct from other people.
Hence why I agreed with the notion above, that a truly eusocial species, one with a "hive mind" or autonomous but not independently-thinking drones or the like, would be one of the few ways to dodge the above--because such entities would lack at least two and possibly all three of the above qualities (they might or might not be sapient, but they certainly wouldn't have self-determination nor personal identity.)
If Dwarves possess sapience, self-determination, and personal identities, why is it their species is what makes them special? Doesn't that erase their individuality worse than the other way around?
Again: no it's not. Because you had to patch something in at the end, after everything was said and done, because the combination of those things would be broken. You had to ban something. That's--by definition--after the problem ("A wizard with too much health would be broken") has already happened. A Dwarf Wizard would be broken, therefore you are now forced to ban them. Cutting the problem off at the pass would be asking either, "How can we make Dwarves still be really robust--important for them in several other classes--without making Dwarf Wizards overpowered?" Or, though I don't think this would be as effective, "How can we make the Wizard generally balanced around physical frailty when some races bring strong (even, potentially, extreme) natural robustness that could completely eclipse that?"
As an example of the latter, consider the 13A Necromancer. One of its class features actually punishes you for having a positive Con modifier (and, if you invest feats into it later on, you can actually get bonuses for having a negative Con mod.) Something like that is a brilliant design move, because it doesn't force players to never play Dwarf Necromancers (for any reason, balance or otherwise), and instead gives them a reason why they shouldn't choose to do that. (or, well, it would in your game's model of stats. 13A stats work differently.)
Er...no it's not. It's a pound of cure. Prevention would be the situation where you don't need to ban anything in the first place. Dropping the banhammer IS the prevention!