1. Shelter is critical to survival. The general rule of thumb places it as a higher priority than food or water. Arguing against people having access to reliable shelter, regardless the rational, is arguing for deliberately killing them.

2. The "they're defective and will destroy whatever they live. Don't let them in!!!" is just calling them cockroaches in a different way. It's fear mongering nonsense and there is no evidence to support that claim.

3. You're assuming correlation does not equal causation. It turns out being homeless, even for a relatively short period of time, is devastating to mental health and even if not the root cause (IE genetic predeposition, TBIs, etc.) it can strongly exasperate them and create some nasty co-morbidities.

Being repeatedly assulted and or raided by police, neighborhood vigilantes and other desperate people is an extremely quick path towards PTSD/other general anxiety disorders. The aggressive de-humunization that occurs can be a potent factor in antisocial disorders. Direct health impacts like physical battery, hypo/hyperthermia, illness, etc. can cause more detect brain damage such as TBIs, etc. Schizophrenia is usually fairly treatable, schizophrenia with PTSD amplified paranoia much less so.

There's multiple groups of homeless people.

There's the long term homeless, who often suffer from issues like mental illness, and short term homeless, who usually don't.

High housing prices absolutely causes people to become homeless when they lose their job, become addicted to drugs, etc.

Being homeless is itself traumatic, and exacerbates most issues homeless people have. Affordable housing and giving homeless people an apartment aren't a panacea, but it does prevent a ton of issues for newly homeless people.

Are you familiar with the "Housing first" model? It posits that even for people who need medical or living assistance, having shelter, a bed, a bathroom, a refrigerator, and a permanent address will allow them and whoever is providing support to deal with compounding factors and receive regular visits, Conversely, attempts to treat something like dementia or substance abuse on the street are next to impossible.

is that truly the case, or just a pervasive urban legend?

which studies support this theory?

You can't solve all their problems with just providing housing, but it would some.

One thing I think people fail to see often when considering programs like this is the generational effect. A program to provide people housing might be considered a failure to some people because many may still choose to do drugs, will ruin their apartment, be violent to their neighbors, etc., some honestly valid concerns. But consider the shockwave 60 years down the line, for the next generations.

Homelessness and drug abuse are generational. Think of a person who would have been homeless who has a child. Was mentally ill and didn't take very good of the apartment, but not enough to not raise the child. Despite this, that child now has astronomically better chances at a decent life than if they had been raised on the streets or put into foster care just because they had housing and stability

You continue generation after generation, and though many people will be considered "failures" of programs like this, the rate of them continues to decrease because the success stories are now out of the system, out of the cycle.

The problem is half measures, which is what we have today. Bandaid fixes that don't get to the root of the issues homeless people deal with, keeping them in the cycle but doing... Something? So they can say look we care...