Sounds v stressful for you all. I don’t think you should ask him to leave.
He doesn’t sound untypical of a teenager with nowhere to be. Lots of them are nocturnal and messy at that age. Going to the gym is a v positive sign and shows he wants to be productive, look after himself and get out of the house. I’d definitely keep encouraging and supporting that. Also a good way to meet new friends and hear of opportunities.
Difficult to suggest solutions without knowing more about his life events (not asking to share, as you shouldn’t breach his privacy). But there are lots of young people struggling right now due to the impacts of the pandemic. He would have been 15 when it started. That’s a tough age generally (lots of angst and hormone turmoil) and the age kids usually start to be more independent, move away from family life and go out with friends. Difficult to know how it may have affected him, especially if you weren’t in the UK: how severe the lockdowns were etc.
What had his life been like up to that point? Try to walk yourself through his life: when siblings were born, when you split with his dad and what led up to and followed that, any moves of home or school, any relationships you’ve had since, his relationships with extended family, family friends and friends his own age. Try to imagine how those things may have affected him. If he’s the eldest, does he feel extra responsibility to you and his younger siblings?
Sounds like he’s stuck in a rut. IME constantly suggesting solutions doesn’t help young adults. It can add to the pressure and can be counterproductive. They have to be allowed to figure it out, (and feel they’re doing it) for themselves. Likewise, pushing mental health or psychological support too much. They need normalcy, fun opportunities and routine to build connections and find purpose.
Do you have support from family and friends who can help guide him? Does he have any strong male role models in his life? If not, is there anyone who may be willing to step into that role for a bit? Do you try to have fun at home, with him involved? Games evenings, movie nights, cooking together, walks? Days out? Does he see any friends regularly? Could you support that? Is there anyone who might be able to ask him to help them out with jobs, even if they can’t pay him? Painting someone’s house? Helping them with DIY? If there is, ask them to ask him directly, and make it clear he’d be doing them a favour because they could do with a hand, rather than because they want to help him. Don’t get involved or let on you’ve suggested it. Otherwise, he’ll feel patronised or you’re interfering and may refuse.
What you should set clear boundaries about, though, is his responsibility at home. Calmly let him know how you feel when you walk through the door to a load of mess. Ask him if he thinks it’s fair he doesn’t pull his weight with household chores when you’re a busy working mum. Ask if he’d cook dinner. Ask if he could do certain tasks. And thank him very much for doing them. It might gradually build his responsibility and pride.
Try to focus on the positives. He’s alive, physically healthy and going through a rut somewhere safe, where you can keep an eye on him and support him. Try to focus on yourself and your other DC. Sounds like you have a busy life. Try not to take your stress out on him. Look after yourself. He’ll sort himself out eventually if you take the pressure off.