Seriously, your not going to become master of the Mages guild if you cant even cast magic.
The mages guild quests in Oblivion didnt even require you to cast magic. It was just the fighters guild all over again.
They influenced you to use magic a lot though, they gave you a staff, mage's robes, sleeping quarters with a bunch of mages, some free spells, lots of enemies that have a weakness to fire, lots of enemies that can't be hurt by normal weapons.
I mean, they can't just scream "No swords!", what about Spellswords, Knights, Sorcerers, Battlemages, Witch-hunters, and such?
I completely understand what you're saying, I became the Archmage on my first account using only my trusty bow. If they give the majority of enemies "reflect damage" in the next game around 20-30%, that'll influence people even more to use magic. However, being a mage has a lot of advantages, including the use of Alchemy to cure paralysis, poison, illnesses, and other things.
They can't just act like: "You aren't skilled enough in Destruction.", if you're a Healer. What I think might work is if in the next game you had to do something spell-related to pass. For instance, like how the Thieves Guild had us running around collecting money, what if you had to cast X amount of spells effectively? They don't have to be super-spells, but enough so that our skill in that will go up.
And they could give us a number of tasks we could choose to do; or they could be randomly generated. Like healing wounded soldiers, ghost-busting, teaching lower-level members how to cast a spell by showing them, reading books, creating potions and poisons, assisting somebody in a task like using telekinesis on a heavy object, giving somebody water-breathing, finding a lost kid in a cave (Light, Detect Life, Night Eye), getting an animal back to its cage without hurting it (Command Creature).
You name it, there's probably a small task you could do with it.