The problem is overlap. That is, if a wizard is assumed to be a specialist in necromancy, in divination, in conjuration, etc., It's assumed they would have access to the best spells for that school. They would have important key spells at low level and the most powerful specialty spells at high level. This creates two problems: 1.) The wizard spell list inflates to give eight specialists worth of magic to use, and 2.) The wizard as built gets access to all spells on the wizard's list, meaning you can pick up the signature spells of other specialists at no cost and always have the best spells.
Neither of those is a problem. Both are good things.
If you really want to narrow things, why not simply make some of those spells be
bonus spells for that tradition, rather than general Wizard spells? Some spells of each school are iconic wizard spells, and wizards in general should have access to spells from all schools, but there's no reason you can't make the spells that make undead servants into bonus "always prepared, doesn't count against number of prepared spells" spells for the Necromancer. Combined with the subclass features of the Necromancer, that should make them great Necromancers. The fact that you can also make a Death Cleric or a Warlock of the Undead patron that is also a good necromancer is irrelevant. Having multiple expressions of "necromancer" that work differently and have different benefits is a good thing.
In AD&D 1e, magic users and illusionists had separate spell lists and the illusionist had spells the wizard couldn't cast, and vice versa. In 2e, they all got lumped in the "wizard" spell list, meaning a mage (nonspecialist) could choose freely from both lists and pick the best of both classes. In 5e, there are very few spells that sorcerer, bard, or warlock exclusively get because the magic they tend to use (charms, blasting, creepy) are also the domain of enchanter, evoker, and necromancer wizards. So the wizard gets the best of all classes, excluding a few signature spells like eldritch blast, chaos bolt or viscous mockery.
Okay. That isn't a problem of wizards having access to spells from all schools, it's a problem of some classes not having unique spells that should have them. Easily fixed. Meanwhile, the spell list is far from the whole of those classes. Most of their flavor and how they play comes from class features. A sorcerer and wizard with the same spells won't play the same, and sorcerer is the most similar to wizard of the full casters.
(I should note here bards being able to heal is an aberration to this discussion: bards are unique that they are the only "arcane" class with healing magic. While this isn't a role the wizard can step into, it's hardly unique with the cleric and druid both being healers. Focusing on the non healing magic of the bard shows they aren't doing much more than a enchantment or illusions focused wizard can do.)
Bards aren't really arcane, they just get called that for some reason. And no, I'll not support stifling the Bard into two schools of magic just so their spell list can be more distinct from the Wizard's. They have a whole suite of class features and subclasses that already make them quite distinct.
I'm looking to make it that the other arcane casters aren't runner-ups to their own types of magic because the wizard has to be good at everything. Wizards are too versatile, and I'd like to see greater balance placed between casters by breaking up the wizard's role as default specialist.
The other casters aren't defined primarily by their spell list, but by class features and subclasses. To the extent that what you're describing is even a problem, it's a problem with
those classes, not with the wizard.
Each class should get new spells that are tailored specifically to that class, that no one else can ever get, not even via a feature that would normally let them take a spell from another class.
Sorcerer should have spells built with more change in how they work when upcast, that take raw elemental power or even raw magic and bend it to their will. Combining that with Sorcery Points will make a class that is comparably versatile to the wizard, but in avery different way. So, stuff like "if you cast this spell using a spell slot of 3rd level or higher, it no longer requires concentration." Stuff other spells just don't get to do. That, and some of those spells should be specific to an Origin, and only that origin can have that spell. Literally no one else can has, ever.
Bards should have spells that resemble the 3.5 Bard's Songs, that start with a bonus action and last at least 1 minute but an hour would be better. Give them a tag that isn't concentration that makes them mutually exclusive. They should be reduced in their focus on illusions, and to a lesser extent, enchantments, given curses and other debuffs, and basically should run the "spells that are meant to represent things people in stories did by simply speaking" game.
Warlocks should get more types of curses, probably sharing some with the Bard but each having unique ones as well, and should have a class feature that triggers when a target has been cursed by the warlock (there is an invocation that does this already, though it may not have survived UA), and should get some of the spells related to invocation and evocation in the RL western mystery cult usage, that is possession and calling forth and binding of spirits.
The non-arcane casters are easier, but I'd also extend this mindset to the 1/3 arcane casters.
Arcane Trickster and Eldritch Knight should both get spellbooks and prepared spells, and "always prepared" spells, with at least 1 per spell level (that they get) that is unique to them.
What
shouldn't happen, is making the wizard more narrow, except as part of supporting the school specialists, by making some of that school's spells always prepared bonus spells for that tradition. If you're an abjurist, you always have Mage Armor, Dispel Magic, and a few others, prepared.
I'd also make Counterspell into a feature built into the basic spellcasting rules. If you have spell slots, you can counterspell. It always requires an ability check, with a bonus equal to +1/spell slot level used to counter the spell. A ninth level counterspell against a ninth level spell can still fail, it just isn't likely, but against a low level spell it can't fail. Perhaps include a rule that a natural 1 always fails, and a natural 20 always succeeds, as with attacks, maybe not.
Frankly, this would help other spellcasters keep up with wizards. I might also add something that if you use your reaction to push back against a counterspell, and you win, the counterspeller takes force damage from the magical energy being turned back against them, equal to your spellcasting modifier plus the level of spell they were trying to counter. Make it risky.
