Morkus from Orkus
That's a lot of twisting to avoid educated wizards. A scholar is a scholar. You aren't a scholar unless you are educated, formally or informally. And a sage is a learned(educated) man, and I agree that a wizard is as likely to be educated as have low strength. More likely really. And you don't get to lecture in a library without being educated to some degree, formally or informally.An apprenticeship itself implies a trade, not a traditional academic education. I could see where you would draw the connection between "education", "study" and "scholar", but that is not how I interpret these and I think those terms are meant to be broader than just an education IMO .... as would be said about someone who is a football player and "studies" film and is a "scholar" of the game which does not necessarily mean a formal education.
Bottom line is I don't interpret scholar and study the same way you do. I don't see this as limiting in terms of play and I see magic itself as an applied field of study for a Wizard, not necessarily an academic one.
I will also note your interpretation would pretty much eliminate Wizards as a viable multiclass option for anyone who was not educated prior to the start of adventuring.
..."likely to come to an ordinary life"... is a stereotype, prefaced by "likely" which by definition means there are exceptions. Further a similar diclaimer about lifestyle could be said for most classes. Saying a Wizard is likely to be a Sage is the same as saying a Wizard is likely to have a low strength or Elves are likely to be thin or Dwarves are likely to have a beard. It is all true in the gameworld, but all meaningless mechanically as I can take my wizard and put all my ASIs into Strength or play a fat Elf (like Mr. Witch from Witchlight) or a clean-shaven Dwarf and be totally within the rules.
The key difference mechanically is a Wizard is "likely" to fit into this Sage mold, where a Paladin MUST have an oath and a Warlock MUST have a pact and patron.
Then there's Xanathar's.
"WIZARDRY REQUIRES UNDERSTANDING. THE KNOWLEDGE of how and why magic works, and our efforts to broaden that understanding, have brought about the key advances in civilization over the centuries."
I mean, that's education.
"The price that wizards pay for their mastery is that most valuable of commodities: time. It takes years of study, instruction , and experimentation to learn how to harness magical energy and carry spells around in one's own mind."
More about the long study time wizards take to educate themselves.
You are correct that there is a disconnect where multiclass wizards are concerned. That disconnect is with allow it at all. They did so in order not to gimp people that wanted to multiclass into wizard as all classes should be able to do so. However, that does not invalidate the explicit call out to how educated wizards are and how long it takes to become one. The most obvious reasoning is that the PC is some sort of wizard savant and achieves the required level of education very quickly, not that education is not a part of the process.