The only way i know it was done closely to your idea officially was in 2e when you were prohibited to cast or learn or even use scrolls of the opposing school but you were not limited to your fav. school. You had lowered chances to learn spells not of your school though (and increased when they matched your school)
The big problem I do see in 5e, is that the available spells are to clustered. So specializing in a single school could give you problems, e.g. if there are some spell levels, in which only very few spells are available.
Also, not every school has all base features that the wizard uses to compensate for his lack of armor, HP and weapon skill available in a balanced way.
I rather recommend using the same methods I suggested in another recent thread: If you want some extra challenge for your wizard, introduce a nice no-magic or wild-magic zone here and there. (Dead or malfunctioning mythals, broken teleport gates, remnants of the spellplague, dimensional rifts, overlays of feywild or any strongly chaotic aligned plane, all of that works as an in game justification)