I spent way too much time trying to track this down in manga, LN and anime. ... It made my head hurt.
So, just the anime... In ep 1.11 he says that he's going there specifically to learn about various things about the world. And later (ep 2.4) he learns that he isn't registering to be a student, but rather as a teacher. So he did go there in order to attend the academy -- which require a huge time investment. Then that... just doesn't happen and he ends up teaching instead. So, going to school to learn just got dropped because the author randomly changed their mind after already starting on the arc.
The anime then gives an additional reason (other than studying) as: in order to open a store, he has to be a resident of the city, which means that he has to enter the academy. ... ... This is just absurd. It's the anime trying to explain away the change from "Going to school to learn some stuff" to "I'm going to become a teacher".
As for the actual, literary reason: Because it's Japan, and they're freaking obsessed with going to high school in fiction. So of course, the author throws in an "academy" arc.
The core of the story is "become an OP lord" -- but that plot point should greatly reduces potential randomness because Makoto is now the lord of the land and should be taking responsibility for that. ...
Now he could be written as a character that doesn't want that responsibility -- except he never says that. He does accept (to some degree) being lord of the demi-plane, so that should require that he take on responsibility, but he doesn't really do so. So he ends up in this shitty status of neither accepting nor rejecting responsibility.
But then in contrast he does accept responsibility for teaching and guiding his students in class -- something that he does spend a lot of time and attention on. ...
Now maybe the author could be writing him as a contradictory, troublesome personality... except the LN never raises that point either. It's just random.
In short, Tsukimichi is not a "wandering character" story; it's an "author throws random stuff in for no in-universe logical reason" story. It's a pastiche of fantasy tropes.