The elephant in the room is, of course, that magic-users aren't supposed to be able to restore damage. That's been part of the game since the earliest published editions where we had Fighting Men, Clerics, and Magic-Users, though there was the occasional workaround. If they could, well, what's the point of clerics?
Now house rules are house rules, and you can make any rule you want; the line between abjuration or alteration and evocation has been blurry on quite a few occasions (Fire Shield comes to mind) and even back in 1st ed quite a few spells were listed in 2 schools.
2nd and 3rd ed put the clerical spells (including of course healing) into spheres (2e) or domains (3e), so you didn't have to worry--healing spells were in the healing domain, along with Plant, Animal, Combat, Guardian, and whatever else you wanted to make up. It's like asking whether an orange is a car or a truck--it's an orange, it's another class of object entirely.
So can arcane casters take restoration magic as a school; if they can, what do they have to give up to get it? Evocation magic would be my first thought--no more fireball--but it's ultimately up to whatever you want to do with your game.