I think that conclusion is a bit simplistic.
I wouldn't consider the huge number of homeless who are drug addicts and alcoholics to be necessarily all be mentally ill. Nor do I think that those who have been kicked out of their homes because they couldn't afford to pay rent all be mentally ill either. All druggies and alcoholics don't necessarily belong in a mental hospital, and those addicts that resort to crime to feed their habit belong in jail, which will dry them out regardless of their addiction, if only they'd be arrested and prosecuted which would get them off the street. The homeless vets should be the first group that is helped with housing provided by the VA, and I'm sure many of them have mental problems as well, and the VA hospitals should step up to care for them. Any of the remaining homeless who are illegal aliens should be immediately deported.
Those few policy changes alone would probably account for 75% of the homeless, which I think is a very good start.