Wait, what? That's precisely opposite of what your source [1] says:
“70% were receiving mental health treatment or had in the past.”
"An April 2016 survey of New York City’s homeless population reported that unsheltered homeless individuals were most likely to be severely mentally ill single males."
Something like 1 in 5 of the homeless in San Francisco have a traumatic brain injury.
None of these people are going to be fixed with mere "housing".
Even worse, putting these people who desperately need medical treatment in "mere housing" is very likely to cause the "mere housing" program to fail when it could have succeeded. The homeless who need "mere housing" don't want to be near the homeless who need "significant medical treatment" any more than anybody else does.
Homelessness has an "Amdahl's Law" nature to it. You have to separate out the different types of homelessness and apply the correct solution. And you will only gain the improvement for the group you "solved".
Consequently, you can solve 20% of the homeless problem and people will still say you "failed" because 80% of the homeless are still in their vision.