I think that the specialist should work like the 4e mage.
Not giving up the ability to use a certain school but instead getting a bonus when using spells of the specialised school.
I prefer to build upwards.
there could be a ability that generalist could get that shows the versatility of the generalist.
From 3e and 2e, for the most part not being a Specialist Wizard was a substandard choice. Some of the splatbooks tried to make Generalists more appealing, but for the most part specialist wizards were generally better.
The effectiveness needs to come from smart/creative play, not from some statistical exercise on the part of the developer/player. Even in 2e, I played a wizard who was, essentially, a transmuter focused on the Item spell and some higher level variants. I had 15+ years of experience vs. the 1-2 years of the artillery wizard, but I was considered significantly more valuable. Even at 12th level, I still had to pull out my crossbow or dagger to actually do damage.Combat isn't just damage.
If your illusion causes enemies to cower in fear or attack illusory targets it can be just as effective as fire.
Ditto.I realized YMMV and all, but my experience is opposite; specialists were rare because PCs rather have access to the best of all schools than get hamstrung by the loss of 2 or more schools. I honestly rarely saw anything but mages/generalists in 2e or 3e
Here's a crazy idea - all wizards should be specialists (in the vein of 3e's warmage beguiler, dread-nec etc.), no single wizard should be allowed to have access to the entire spell-list that allows them to do everything.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.