Why can't these be subclasses? Or builds? What about necromancers? Or enchanters?
I think you're fundamentally on the right track here, but I think saying that people who like wizards fall into Blow Things Up and Trick Things camps is missing the mark a bit.
So, I think you're right in that the "wizard" class in D&D is hell of broad. Not coincidentally, the "Fighter" class and the "Rogue" class (and, to a lesser degree, the "Cleric" class) all have this problem.
Is your fighter a knightly defender? A reckless guy with a big weapon? A brilliant strategist?
Is your rogue a thief? A spy? An assassin?
Is your cleric a healer? A crusader? What if they serve a god of thieves and trickery and darkness?
Is your wizard a boom-maker? A trickster? A scholar?
When I look at the spell list in OD&D, when wizards were at their most limited, I see charms and transformations as the main themes: enchantment and transmutation. It's not monolithic (there's some booms and some illusions and others), but those are strong themes. This is a bit of an awkward fit for the Timmy/Johnny/Spike scheme, but then Timmy, Johnny, and Spike are three flavors of fiero, and "to win" isn't the only reason people play a particular class, so it's not I think the most useful way of looking at D&D class structure.
Which is part of where I think you're a little off base. A lot of people play wizards because they want to pretend to be a wizard, to do things that they imagine wizards doing. There's a lot of things that wizards do in various media, because "it's magic" is universal fantasy phlebotnium, and so D&D wizards end up being capable of doing anything. Which makes them hell of broad.
But I don't think the solution is two different classes, any more than I think the rogue needs to be split into a "thief" class and an "assassin" class and a "skirmisher" class. Rather than to divide it all up, I see embracing the multitude as a pretty good idea, because it enables more modularity: someone can be a little bit thief and a little bit assassin; someone can be a little bit illusionist, and a little bit evoker. With one Wizard class, you can play an evoker or a transmuter or an enchanter or an illusionist or a summoner or a runecaster or a necromancer or some sloshy hybrid, and you don't need to silo them in their own little towers.
I think the "problem" of versatility here should be broken down more effectively and specifically. Versatility isn't itself a problem -- being able to fly, shoot fireballs, and turn invisible isn't a problem when the party fighter is capable of the same thing (winged sandles, exploding arrows, and a cloak of invisibility!) -- it just becomes something ANY D&D character is capable of, part of the play experience. D&D has more typically had a problem where the wizard's ability to do that is granted and automatic and presumed, but the fighter's ability to do that depends on DM generosity and the correct magic items. That problem can be fixed in either direction: either the DM gets to pick the wizard's spells (something older e's frequently did), or the fighter player gets to pick their own magic items (such as via a crafting system or special mounts or somesuch).
You could also say that NOBODY should fly, turn invisible, and shoot fireballs (a less extreme version of that was a part of 4e's solution). Not the most satisfying in my mind.
I think siloing class abilities of a wizard into unique spell lists isn't really the answer, here. Invalidating the concept of a generalist wizard (and, in the process of which "classes" you select, the transmuter, the enchanter, the necromancer, the summoner...) isn't the best idea for D&D, I think. There should be a generalist wizard, the generalist wizard should just be balanced with the other classes -- they don't get to be the only ones capable of flying, turning invisible, and shooting fireballs. OR, those abilities aren't nearly as good as they have been in the past.
Now, in my personal game, I'm playing around with getting rid of all generalists (no rogue, no fighter, no wizard, no cleric) and concentrating on specialists (thief/assassin/spy, knight/slayer/cavalier, transmuter/necromancer/illusionist, protector/gish/monk), just 'cuz I kind of want to see what that looks like. But even there, I'm not making the distinction at the "class" level, and it's not meant to control the arcane/martial divide. All of these specializations are going to be "power source neutral": an arcane assassin and a divine assassin and a primal assassin all are going to have the assassin's abilities.
One of the things I'm realizing in doing this is that there's a LOT of specializations...