That is simply not true. Homelessness goes up and down with housing prices. You can see that in the data.

If housing is cheap, mentally sick person or even addict can get one in an easier way. And their relatives have easier time to help their mentally ill relative find a place.

Being homeless makes both addictions and mental illness worst, to the point of unsulvable.

But it also affects your safety net. If rent is 60% of your income, it's much easier to drown in it than if it's 30%. Especially considering once you start drowning, you'll need to pull 3 months' rent out of somewhere for the deposit of a new (cheaper) place.

It also makes it make more sense to live in a car. A car costs the same, mostly everywhere, so if you're in a stupidly expensive location, it might start to sound reasonable to live in a van.

Drugs, poor financial literacy and insanity are the main reasons, but expensive housing is also there.