If I were to do an alternative magic system for 5e, I wouldn't want to make it compatible with the existing classes, and at this point I'd probably end up scrapping a bunch of core rules to implement house rules I like.
For instance, spells per day don't interest me. I'd have some base mana that you start every encounter with, and then different ways based on magical tradition to get extra mana.
Like a pyromancer might have a 'cantrip' he can cast at will that creates a shell of fire around him as he draws upon the flames that fuel his magic. It might burn enemies a bit, but more importantly it charges up his fireball on the following turn. A mid-level pyromancer would start each encounter with enough mana to cast a couple strong spells, but after that he'd have to go back to drawing mana.
Or a necromancer might be able to suck a little life with his hands, but if he wants to cast something powerful he'd need to slit the throat of a dying creature.
Or a member of the Wayfarer's Guild could get mana by dashing, drawing upon their own movement to kickstart spells.
Different magic traditions would yield different play styles.