Just not true.
If you include the invisible homeless in the group, you can probably say that the majority of the homeless don't have mental health issues. It takes functional mental skill to not be a visible homeless person.
The highly visible homeless probably have significant mental health issues. Mental health issues have a nasty way of completely destroying your social network. If someone doesn't get medical help before they hit bottom, they're in real trouble.
The biggest problem is that the "homeless" aren't a monolithic bloc. So, Amdahl's Law is at work--even if you completely eliminate one class of homelessness--you still have 80% of the problem left, it looks like you didn't do anything and everybody gets angry at the waste of money.
You both can and must address everything in parallel to actually dent the "homeless problem".
That's the point.
Let's say there are 5 primary classes of homeless. Even if you take one of them and reduce it to zero (and you will never accomplish this)--you're still left with 80% of the problem.
Only now the people approving the checks are angry because the "homeless problem" doesn't really look any better.
Anyway, I agree with your general sentiment, and it tickles my mind. Thank you!