There are absolutely some homeless people who choose to reject housing assistance and shelter programs. I know because I did some volunteering to help the homeless and met a handful of individuals that were happy to accept donations but refused to be committed to any program. They refused to have rules imposed on them by the shelters and preferred the freedom of doing as they please. One of these individuals has been homeless since the late 80's and seems quite content with his lifestyle. I'm not claiming this is typical, but there are certainly more people like him out there.

Many homeless reject shelters because they would have to give up much of their personal possessions, can't have a pet, live under tight restrictions, and deal with theft from other homeless people in the shelter.

Giving people small, secure apartments would solve many of these issues along with many emergency hospital visits and many other issues.

Many shelters in the US are run be religious organizations and discriminate against gay people, too. This is particularly a problem because LGBTQ make up a sizeable chunk of homeless youth (think: kids getting kicked out by parents).

You really can't imagine why a person kicked out by their religious parents for being gay would be apprehensive about entering a homeless shelter operated by that same religious group?

That would be quite the coincidence... and also assumes there is only one shelter with free beds.

I find it almost incomprehensible the first time sometimes told me (in the UK) they were going to sleep on the street because the shelter didn't allow drugs. I don't think they were a particularly special case.

Victims of circumstance to some extent, but not stupid nor really living their preferred life I'd wager.

I suspect you need a stable caring society for a couple of generations and still there will be some outliers. A worthy cause to address though.