I'm not following. Here's the claim, as I interpreted it:
"Many homeless people do not want shelter because they are mentally ill."
Here's how I responded: first, there just aren't that many mentally unwell homeless people in the "naked and screaming" sense. Second, that those who are mentally unwell, to whatever degree, do not accept forms of shelter not because of that illness but because those forms of shelter are, by normal standards, extremely degrading. We would never dream of asking someone who hasn't otherwise been dehumanized to willingly subject themselves to constant surveillance, a living space shared with desperate strangers, and the complete absence of any guarantees around the security of their private property.
Put another way: being homeless doesn't make you lose your sense of dignity. Being mentally ill doesn't either, except in the worse cases. Treating the overwhelming majority like they're criminal timebombs is dehumanizing and just doesn't match the facts on the ground.
So again: what's being missed here? I'm interested in the 99% case, which includes a very large number of mentally ill homeless people. I'm not engaging in the 1% case, because I think it's a frivolous diversion from the needs of a great many suffering people.