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.

That doesn't apply to the chronically homeless. The majority of homeless people at any given time are 'down on their luck' and relatively quickly get out of homelessness. You also have to factor in the role drug use and mental illness play into homelessness. Many of the chronically homeless fit into one of those two categories, and a subset of those people do not want or will not accept help.

Coincidentally, I just posted a piece two days ago that spent substantial time on the front page about people who grow up in upper class circumstances and wind up on the street at some point:

https://news.ycombinator.com/item?id=25310779