Give him credit for admitting he has a mental illness. There are too many people that won’t do that much.
Why would things be any different this time without dealing with the underlying issues? I’m not judging him any more than I would judge someone with cancer. Mental illness is just as real and could benefit from treatment just like any other illness.
No, you aren't judging him. You are merely condemning him to a no win situation. This is done all too often to both mentally ill people and homeless people.
What would be different is that I have given him a list of remote platforms (plus other supporting information sources). If you are homeless but can access the internet, you can do work on the internet and no one cares when your last shower was.
Holding down a job while homeless is a tall order. Some people do it, but it's hard. Doing remote gig work online is very do-able. I did it for years while working on solving my personal problems.
I was frequently a loon while homeless. Doing remote gigs meant that I just didn't work when I was a loon, and I wasn't fired for taking time off. As long as I completed gigs in a timely fashion when I agreed to them, I got paid and I built my skills and reputation and increased my hourly pay and it helped me eventually get off the street.
Best of luck in finding solutions that work for you. :)