There ought to be two kinds of spells (no spell schools): ritual and psychic (intelligence and will). Ritual magic can be cast by anyone with the right ingredients, amulets, trinkets, etc. However, if the ritual spell is incanted improperly, or by someone of low intellect (gone), the result can be disastrous.
Psychic spells are the result the PC is "imagining" (equip the spell, to represent the caster's occupation with the thought). These spells gauge the caster's will (gone) for their effect/s. They're always successful, and these spells you only need to learn once. Ritual spells require new procedures for the better effects.
Also, spells are state secrets; they should be guarded very closely. By the fourth era, there should be no spells in the market place. All wizards must register newly discovered effects and efficient procedures, with the nearest Guild. Failure to do so can result in stigmatization. Tell me what you think.
You can do something similar by keeping the spell schools and making the intelligence spells all be ritual magic. Which they likely are already. I forget the book, but one described conjuration as a ritual these witches performed to summon daedra. Potion brewing and enchanting are pretty ritualistic already, and the former mysticism could have adapted well enough. They should require your grimoire.
Your psychic spells pretty much already describe your willpower spells.
Spells aren't quite state secrets already, but magic is already fairly controlled under the Mages guild. You recall from quests in the past three games that they order the deaths of anyone practicing magic they don't approve of or aren't getting a cut from.