It would cost, but you can externally monitor and correct institutions if you have entirely separate people doing it.
Or - livestream everything and let the public monitor it. (Obviously this can't be done for various reasons, but we now have the technology to literally record it all.)
Shuffling mumblers are not dangerous.
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.
My position is: if you're a danger to yourself and society you should be coerced into receiving treatment (either in a prison or mental institution) if you are not you should receive housing and non-coercive treatment options through a publicly funded healthcare system.
This seems weirdly controversial though. Seems like people vehemently opposed one or the other half of the solution and we get the terrible situation we have now.