The one place I know of where a housing-first policy has been implemented, Salt Lake City (or maybe all of Utah?), doesn't just give homeless people a set of keys. They do assign everyone a social worker at the same time. The goal is to minimize the risk of what happened to your friend; that isn't good for anyone, including the person spiraling.
It's key, though, to start with a home (and a social worker). The wealthiest country in the world can do that for its people -- and it shouldn't be on private citizens, though that was generous of you.
My ex was schizo-affective (had traits of both schizophrenia and bipolar disorder). He was on disability. His doctors did not want him living by himself. Some things were simply too stressful and he couldn't do all of it. He'd only cook instant food. He wasn't good at budgeting money. And so on. For a while after we split up, he had a social worker coming to his house for wellness checks. My friend worked a similar job for another hospital: SHe'd visit mentally ill patients and help them, sometimes shopping and whatnot.
There was no recovery for him, only meds. That's it. Only meds.
I cannot fathom how stressful it would be to have your housing and your safe space be dependent on taking your meds, especially when your meds need adjustment... and paranoia strikes up again. Every stressor simply complicates things.
Threat of homelessness is basically the only remaining lever the law allows in these situations.
The other two points are spot on.