You're mixing two different things in your argumentation: what can theoretically happen, and what typically happens. Majority of people have to completely alienate their friends and family with their asocial behaviour to become homeless.

I've only been technically homeless and only slept on the street once in my entire life, but I have some time volunteering, talking and interacting with them, and in my personal experience, overwhelming majority of them are not just normal people who felt on hard times. I'm completely sure that if they took rational decisions and controlled their impulses, majority of them would not be where they are.

However, and this is a very important point: division between mental illness and bad decisions is very murky, so I'm not making any moral judgements here. But if a person suffers from a mental illness which leads him to make irrational decision, it is a tragedy, and this person is not at fault, but society doesn't become responsible for this awful situation either. It is moral and commendable to help these people — that's why I do it, for example. But there's a pretty huge gap between doing something out of kindness and having an obligation to.