Yep, I agree there should be a generalist. Along with beefing up the specialists a bit to compensate for the lack of flexibility.
I just think it's an easier solution than a custom spell list. Since it would also limit cantrips, it seems like a simple answer.
Fair enough. It's not a bad idea, to be clear; I just wanted to point out the obvious-to-me objection (which is easily solved, once seen).
As for me, I don't see much of a way to make the spellcasting classes different from each other that wouldn't make the mechanics more complicated. Since much of what I like about 5E is the way it's streamlined, it's not a thing that seems worth the doing to me; obviously, people will have different preferences.