As for the “suddenly, I’m a wizard too”, that’s not much of problem for me, as I can assume that the desire/knowledge/practice has been happening in downtime that hasn’t been referenced in the game until it now became important. I mean, I don’t normally roleplay out the fighter doing his nightly practice or the cleric doing evening prayers and reading sacred texts for those that began their careers in that class - but I can assume that they are doing such things during the times between adventures.
I wonder if part of it for me is trying to imagine what skill level the 1st level character is supposed to be? Are they an apprentice, for example, or a journeyman?
If 1st level is apprentice level skills, should stuff like a first level character has be pretty easy for anyone to get and be common? If 1st level is journeyman level skill, then is it something folks pick up on the side that's so easy they don't have to announce they're doing it until they get it?
[In any case, I am all on board with admitting we all have game pet peeves and this happens to be mine, instead of say AC or HP or whatever bothers anyone else.]