To me at least, it seems logical that a reasonable person would choose shelter over being outdoors and injured for 15 years. Making that choice is their right, but let's address their mental health problems first to give them the autonomy to make that choice.
Homeless mentally ill people be lying to save face, or that me be at least a component of what they are saying.
If you spend enough time with them, you may get a lucid moment where they can be honest with themselves and you; but it is very time consuming to get there.
A good comparison is a young child (or an elderly person slowly getting senile)! They often believe silly things, but it's not all or nothing. It's possible to "reason" with them, but it takes more effort, time, and care, that's all.
99% of people with milder forms of mental illness do need to be treated with respect of autonomy. It's the last 1% that worry me.