In thinking more about this, I'm starting to feel like developing solutions specifically for the most mentally ill/unstable individuals is important.
1) They're a danger to everyone including other homeless.
2) They have an outsize impact on people's frustration with and perception of homeless people. Walking down the street and worrying about being accosted by the guy yelling at himself is a powerfully bad experience. Walking by a couple guys in tents or sleeping bags may not be everyone's cup of tea but it's not going to emotionally scar anyone.
3) This segment is the hardest for us to confront on a human level, we want to believe in rehabilitation, that we can help, so the resources (shelters/homes) are designed for the most capable of homeless, kind of ignoring the elephant in the room.
4) Have there been mental health hospitals made specifically for the homeless? It seems that the most mentally ill of them basically need intensive care to even have a shot at some degree of rehabilitation.* Unlike a shelter with services this would be involuntary and they would not be able to leave until meeting some criteria. May be a little draconian but it beats constant interactions with police and going in and out of jail.
5) A lot of the "regular" homeless people I encounter don't really seem that interested in working their way towards a job and a house. They seem fairly content with their lifestyle. With the most viscerally troubling aspect of homelessness being mitigated, maybe we can push more toward normalization: sanction the tent city, provide some light services/development, give them a bit of dignity and the peace of mind of not being a criminal. Seems a lot cheaper and more effective than building shelters or houses and relocating etc. People would probably hang out/sleep in the street/sidewalk less, and those that do could be ushered to the acceptable area instead of the farcical "you can't be here...or anywhere, technically, but just move somewhere else for now."
Does this make any sense? Has stuff like this been tried?
* In general I think this is one of the big issues with homelessness - a lot of them need care more than anything else. As in someone literally holding their hand every day, preventing them from regressing to their habits, and (with infinite patience) constantly reassuring and rebuilding their psyche, until their brain is sufficiently rewired (a timespan measured in years). The problem is that the number of people that could use this kind of care is far greater the number of people that are willing/able to provide it, but developing programs around that could be an effective use of the funding.