There are two types of homeless. The temporary homeless who have been caught out by circumstance and work mightily to avoid remaining homeless. These people tend to be homeless only for very short periods of time (days).
Then there are the chronically homeless, who are generally unable to live within societal norms. Many of these people are mentally ill. And for many of these people giving them a home and money will not solve their problems.
This thinking is based on the fallacy that mental illness is a purely biological condition, which anthropology, epidemiology, and clinical data all soundly disprove. Even for mental illnesses that are mostly biological, like schizophrenia, the quality of outcomes varies enormously depending on the person's social environment as well as the course of treatment.
First thing they'd tell you is that while mental illness isn't biological, that doesn't mean there's a cure. Some people could go 20+ years in therapy and still end up only a slightly less sick person than they were going in.
Also, many mental illnesses become biological in the sense that brain chemistry actually changes.