Not if the government takes the perspective that housing is a human right, regardless of an individual's mental health or substance abuse problems, willingness or capacity to work, or even criminal history. Some would argue that a country has a duty to shelter its citizens, and research shows that the more difficult problems associated with homelessness can be solved by focusing on housing first, then taking on other issues.

The Housing First movement takes this stance, and has seen some success: https://en.m.wikipedia.org/wiki/Housing_First

I've never been homeless, but I do think that if you've been knocked down every time you try to get up, after a point, you'll stop trying to get up... Nor will you take a hand that's being offered.

Most of the help the homeless get is ephemeral, at best - and won't necessarily get them off the street. It's no wonder that many choose to not fight that fight.

It's pretty clear that you have absolutely zero experience about mental health or human psychology.

Amongst other things, most people will attempt to adapt to their situation and retain a sense of agency. That is one reason why you'll find that homeless people speak of the positive aspects of their lifestyle and act as though it's a choice. This is no different from a startup founder extolling the virtues of their 80 hour week.