Yeah, I played the game to the (current) end. I can see how the MC can improve over time, I just find him too whiney. And yeah, he's not a teacher, but a psychologist, but I still don't think there's really any legal issue there. Even ethically, it's not so bad. If the patient needs someone else to play video games with them to keep them faithful to a sleep schedule, getting a friend to be that person would be ideal, but there's nothing wrong with the psychologist taking that role if there isn't really a better option.
Probably, being a friend to the patient could be a good idea, but at least in the story context, he's not worried because of the action, he's worried for go any further with her, don't forget he's pretty new to advising people, so, he doesn't even know what is illegal or not, he's just guessing at that point.