I find it odd and awkward that, still to this day, the only way one can cast 'destruction' and 'restoration' spells are through staves when staves always had the theme of casting one particular spell unique to the staff or a common spell under charges to save a hero's magicka.
I realize tons of spells in the current classes already have similar key spells to those schools (night blade invisibility for illusion, destruction spells spread through all classes, conjuration for daedric summoning, etc.) but I think there are still plenty of options in all the schools to get over the redundancy.
For example, mysticism could have telekinetic and absorb stat/damage/resource abilities. Even though soul trap is in that school it doesn't need to move there. Illusion could bring back calm/fear/rally/fury, and bring back chameleon or muffle rather than just giving a pure invisibility skill for all classes. IMO they should probably just make destro and resto abilities into their selective school, rework staves to hold spell charges, and give an option to make some weapons scale off magicka (or make staves melee weapons but idk, I just don't see why magicka users can't use basic attacks without it *** over their dps). Alteration can use shielding spells, burden (probably not feather unless it only held pve use or gave major/minor expedition), and jump/levitation spells if they were properly balanced (such as being unable to cast spells during the duration of levitate, or having it have a very short duration with high magicka cost). Conjuration could have all the daedric minions that were left out of sorcerer, raise zombie/skeletons, and bound weapons.
I kind of just wrote this in hastily and in spurts, and this isn't really a "you need to take this suggestion seriously!" thread, but at the same time I'd like to hear what others think or see if they have any specific ideas themselves.