Right, which means it seems wrong to incarcerate them in a mental facility against their will. I'm assuming very few would willingly be incarcerated.

Maybe there's a way to house them in some kind of voluntary mental facility only for non-dangerous mentally ill people. They can come and go during the day. But at least they have a roof over their heads, someone to look after them, provide clean clothes, etc.

Feels like someone must have already tried this and none of the mentally ill homeless were interested.