Is saying that many of the homeless in America are mentally ill blaming them? It's our health system that can't handle them, for a plethora of reasons. It's not the homeless peoples' fault for having schizophrenia or PTSD or whatever mental illness, which many of them do.
Homelessness can't be fixed "at source". People become homeless simply by losing their job and defaulting on their mortgage.
Walking around San Francisco you would get the impression that most homeless people are mentally ill, but you'd be missing all the people sleeping away from busy streets, in tents, in their cars, squatting in unused buildings, couch surfing between friends, etc.
Poverty isn't caused by mental illness though it may be correlated with it. How many were mentally ill before becoming homeless? At what rate would wealthy people develop mental illness if you force them into the same situation?