Cheap housing isn't a panacea, but if there was sub $500 dollar rent in NYC you'd see a lot less homelessness.
On the other hand though, a lot of who’s technically homeless at any given time aren’t that chronic, mostly hopeless, and very visible set. It’s people who did lose their apartment just barely after an unexpected job loss or medical expense. Those ones could be helped by cheaper rent! But that group isn’t very visible, doesn’t bother anyone, and often couch surfs, sleeps in their car and showers at the gym for a couple months, etc. and most importantly, they use the many safety net resources to get back on their feet (getting a job usually).
But for the people on the edge of homelessness, that $500 rent could be the difference that prevents them from going down the death spiral of homelessness, lack of options, drug addiction, etc.
Doing that masks the fact that there are working people with housing insecurity, by putting them in the same category as emotionally disturbed drug addicts.
In every analysis I’ve seen, poor people with housing insecurities outnumber the mentally ill type homeless people by three or four to one. It’s very possible to help them, but very difficult to talk about it because everyone imagines you’re talking about the other “chronic homeless” type.