The fighter wanting to become a mage though... Where did he do his apprenticeship? How long has he been studying musty tomes? If we've learned anything from fantasy literature it's that mage apprentices are taken in young and worked like a dog to learn patience, persistence and a smattering of technique. Where did Joe the Meat Shield learn this?
Being an Apprentice mage at a wizard's school or under a wizard is likely filled with a lot of groundwork and stuff that is not specific to Casting Spells. Spellcraft is specifically identifying schools of magic/specific spells (so how to tell the aura of a conjuration vs. an illusion spell). Arcana is knowledge of various magic-related things like ancient wizards, dead languages, the history of spells, dead monsters, spell component combinations and what monsters make best spell components etc. There are likely classes on how to begin crafting your own spells, how to make spell scrolls (and the basic knowledge of how to build magic items), how to contact outsiders. Additionally you'd have non-magical learning like the fantasy equivalent of science, and wizards know lots of languages (due to high int) - where do you think they
learned at least
3 extra languages? When it finally comes to actual casting, it's likely how to shape a basic illusion, how to do a conjuration correctly. along with lots of fundamental practice.
This is a lot like the real world where going to a 4 year college gives you a lengthy education consisting of a little info in a lot of areas. You have general education forcing you to get a well-rounding of unrelated info. Then, even in your specified field, you get a lot of initial info for many different areas within that large subject. You get very little
depth of knowledge. Worse, you get little hands-on experience.
So very little of Mage School could prepare wizards to be an adventuring spellcaster whose main job is killing things very effectively with combat magic. Meanwhile, a fighter who is multi-classing into wizard is likely doing it for one reason: to learn an effective attack spell or a buff spell. He's trying to cast one or two things, and
that's it. He's looking for something really focused, and he's probably been exposed to enough magic to get the fundamental 'say this do that and hope real hard" that the
basics to casting the lowest of spells.
In the real world, many of your jobs have a narrow focus. Experience of being exposed to things in other fields can give you a taste or some justified experience - but it's not hard to get "on the job training" to be capable of doing something specialized. If all you're doing is handling a database, then you don't need a lot of unnecessary info. You just need to be shown how, and then keep doing that.
A different analogy would be the difference between basic training for enlisted soldiers, and officer's school for officers. Bootcamp teaches you how to shoot, trains your body to deal with harsh conditions, teaches you order following, marching, teaches a few skills, and then you might end up going to the front lines right then. Officer's school sends you to get an
education for several years because you're going into command positions. You'll be in fighting situations sure, and that skillset is there, but you'll be doing
more than your infantry men. The magical equivalent of bootcamp is when someone multi-classes into Wizard.