Honestly, you seem to understand the concept of the Wizard, but you just don’t like it.
My thoughts…
Arcane magic isn’t about studying everything. It’s about studying Magic, or at least what mortals know about it. Magic is inherently mysterious, but we can know a lot stuff such as behaviors, patterns and processes. Studying it means understanding more of this mystery, and the ability to use it to his/her own benefit. They’re basically the guys that study enough to know the cosmic cheat codes one can use. Studying more about the “arcane” (mystery of Magic) is their power.
Arcane magic is also defined by what it is not: It doesn’t come directly from the gods, neither from the Plane of Shadows, neither from your own mind, etc. It’s like a giant pool of magic that can be utilized by wizards.
However, since Magic is essentially weird and irrational, there’s nothing that guarantees wizards have access to all magic. It can be just a property: magic from study gives you these effects, magic you pray to the gods gives you those effects, etc. It simply is. Who said Magic must obey to our silly mortal logic?
I don’t know if I’m being clear here, it just takes a while to explain my full view of magic. However, the resume is: weird, mysterious, cheat codes, different access gives you different powers, and illogical.
I would be against such a change you are mentioning. Depending on the game and setting, magic coming only from specifically someone or something can work well. However, in a “generic” High Fantasy game or simply Old-School-style, I want my bookworm wizard doing awesome stuff without having to make a pact with the Devil or something like that. Other classes are more than welcome, but don’t take out the classics