I'm not sure you're being friend zoned quite as much as you are absolutely being put in the position of being his crutch. He wants to be accountable to someone and he's making that someone YOU but I'm really worried that's going to bite you in the butt later. I can see him saying that you pushed him into certain things, etc., you know? It's not a great scenario given his depression and your habit of control. I don't want him to out you in a place where he could accuse you of trying to strong arm him into anything.
He's putting you in a bad situation. He's clearly codependent and if you continue to be his crutch, you're enabling him.
Be careful of that.
IMO, it is best if your checking on him is VERY limited and done from a place of serious detachment, which given the newness of your sitch, I'm not sure that's possible right now.
Your H sounds like a bit of a manipulator by playing the victim. His parents are covering his living expenses. His work is dealing with his strange hours because of his therapy schedule and you're being sucked in to checking in on him.
Maybe a vet can chime in.
And him checking in on you is him furthering his crutch. He's looking to be cared for but he needs to learn how to do that himself and you have to learn to let him do that.
It's going to be hard for sure but there's power and freedom in that.
