@manlycoffee where I live, a main contributor to homelessness is unaffordable housing prices; certainly, increasing supply would help with that. However, the government has a long and sordid history with public housing, which has historically been appallingly racist in its implementation.
We also have a big problem where homeowners and landlords oppose new construction, thereby increasing the value of their properties while making housing unaffordable for anyone else. If we could overcome that hurdle, I think it would likely go a long way towards reducing homelessness.