I think there is a spectrum of homelessness ranging from the totally feckless to people whipsawed by the housing market to privileged kids getting suspended from Barnard and whining about being homeless to claim some sort of intersectional merit badge. I think the problem needs to be addressed with a complicated spectrum of solutions.
My experience in the field has been that many people are simply not capable of the responsibility of homeownership even with steep subsidies. Public housing can be tricky from both a NIMBY and crime standpoint. Rent control and zoning can serve to warp the housing market in all kinds of ways.
Rhetoric about life, liberty and the pursuit of happiness are all well and good, but the many differing segments of the homeless population will need to be addressed at different levels in many potentially messy ways.