This is the first time I've done this:
TLDR, but in honesty I did read the first page. Cuz i already know what's in pages 2 through 4.
My two septims:
Single player game. Do your thing. Every player starts with a basic healing spell. It is then assumed in the land, that there is some form of basic restoration magic accessible to the entire population. Perhaps only the beggars do not know this. People who live in the Warrens and various other places throughout the land.
With that basic restoration knowledge, I think it's reasonable to assume that there would be quite a few intrepid souls who wish to study more magic. I can't imagine branching off, at least for a warrior with a basic healing spell, into destruction or illusion for example. I can see quaffing ready-made potions, but I can't see the character making their own. But then again, I can, if you're a barbarian, experienced in the flora of your world, I would expect you to be able to brew up poisons and potions, perhaps not a full alchemist, but something....
Interpretation.
Unfortunately the game, Skyrim, in its vision, limits your choices, forcing one type of character, to do other types of things:
Open spells. Have to level lockpick, instead of just using a spell, like a mage would have probably figgered out how to make this spell by now. Oh yeah.
Cannot pay someone else to fix your gear for you.
Cannot pay someone else to recharge your magical weapons. Instead, through the course of your hopefully successful and long life, your warrior becomes a master enchanter. Through no effort and desire except just trying to keep your gear tip-top, as any warrior would so desire.
It's the game limitations that create this situation.