I think specialization needs to go much further than it has in the recent past. And it needs to *not* be based on the spell schools, which are artificial, vague, and never intended to support such use. (Frankly, I think the spell schools as we have known them could just as well disappear. They have just about no mythic or fantasy basis and they have never been used consistently.)
I would never willingly play AD&D again, but one thing it really did get right was the Illusionist, in my opinion. It was a fascinating class, and one I very much enjoyed playing. For one thing, they were not limited to just the Illusion school, despite their name; they had a lot of enchantments too, and a smattering of other things. For another, although there were many things they could not do, there were also things they could do that no "Magic-User" could. Generalist wizards couldn't cast Shadow Conjuration or Phantasmal Killer. When 2e dumped all the wizard spells in one box, it was a huge disappointment to me.
The sort of specialization I'd like to see is thematic, and in that sense I think the OP is sort of on the right track.
I don't think that 'Evocation' is really a useful specialization or concept. But Elementalist, for example, certainly is. As is Necromancer, though not limited to what we currently think of as the 'necromancy' school.
I'd like to see a group either of classes or of subclasses or whatever you want to call them, that have their own distinct spell lists. (With a certain small number of spells common to them all. Detect Magic is an obvious choice.)
And I would like there not to be a generalist option at all. Or if there is, that it be not only jack of all trades but also master of none. A generalist casting illusions should not only cast illusions in an inferior way to a true illusionist, but there should be some illusions that are just plain beyond his comprehension. Each subclass should get its own set of cool toys that others don't get to play with.
I'm not very familiar with the 3.X Beguiler, but I'm told it is an example along the lines of what I'm looking for.
(A bit of a tangent, but I've sometimes wondered if the classic D&D magic-user could be reinterpreted as a specialist wizard of its own; perhaps one using Force magic. Magic Missile, Wall of Force, the Bigby spells, and so on. Or perhaps a Thaumaturge who specializes in flashy transmutations, whiz-bang evocations, and so on.)