It is ok for a class to have a feature that boosts the effect of a specific spell. The design space of the class is adding to the design space of the spell.
The design space of the spells is complex and extremely important to balance well, for the sake of the gaming engine, especially at high levels.
Each spell must be able to stand on its own, and balance well alongside the other spells in the same slot.
It shouldnt matter which spell a player chooses from a slot. They do different things, but are equally desirable, effective, and useful. If one spell is obviously better than the same-slot spells, or obviously worse, it is a design problem.
Disentangling spells from class exclusiveness, helps when monitoring how spells perform during gameplay compared to each other.
Spell bloat is a problem. Bloat drifts into power creep and imbalances between spells.
Again, it is ok if a class has a special feature to augment a spell. For example, the official 2014 Fireball spell does too much damage for its slot. It actually skews the balance of all other damage spells, especially in higher slots.
Slot 3 should deal 6d6 damage. There is math. Damage ≈ 2d6 x slot level. This math keeps higher level gameplay from crashing.
With a bit more nuance, a damage spell that only deals damage and no other significant effects, does well to add an extra +1d6 damage. Thus a balanced Fireball spell can deal 7d6 damage.
But Fireball currently deals 8d6 damage.
It is better to fix the Fireball spell, making it deal 7d6 damage, thus balance well alongside other spells.
Then it is ok if the Wizard Evoker subclass design space has a feature that adds +1d6 fire damage to the Fireball spell to produce this 8d6 fire damage. (Perhaps this feature adds 1d6 to any fire spell.)
Really, there is no such thing as a "class exclusive spell". Either it is a spell or it isnt. If it is a spell, then it must balance tightly alongside other same-slot spells. If so, it doesnt matter which character concepts take the spell. However, if it fails to balance with other spells, then it isnt a spell. It would simply be a class feature that has nothing to do with spell design space.