The types of homeless people who are best helped by "housing first" type aid are not the same ones getting arrested/going to prison. Homeless people aren't some big monolithic group you can throw one solution at and have it work...
Even if you're going to overly simplify things you'd at the very least have two groups; the "entrenched" group (more visible and what people think of when talking about homelessness), and "invisible" group (the ones where the factors causing their homelessness are mostly financial).