I see this question come up on here from time to time. I don't know much about homelessness in general, but I do have a brother who is a little mentally ill. Among other problems, he is incapable of keeping a place to live habitable. He utterly ruins a place over 2-3 years, gets evicted, and finds someone else who will rent to him (which is hard, he is blacklisted).
Something in him just won't train his dog, won't put food away, won't not get just angry enough to break things. Yet he appears like a reasonable person on the street. He spends most of his time at home too, so his life is pretty miserable surrounded by these miserable conditions.
We've tried hundreds of things to help him over the years, since I was a pre-teen, and it seems so intractable to help even just him, even with what should be enough money to support one person. It's hard to imagine how to solve the problem of all the world's homeless.