It was not the cost that killed public housing it was horrible social situation that resulted when you concentrated poor people in public housing. It was a failed experiment. New models for housing assistance are being tested. Most of the homeless need way more than housing, they need treatment, counseling and not just given a home/apartment and be expected to keep it up. Shelters seem to be the best model for the homeless. The poor need financial assistance for many things including housing but stuffing a concentration of poor into a large housing block will lead to worse outcome.
If putting poor people in large modest apartment buildings turned into a nightmare, imagine if you have the poor squatting in abandoned or vacant buildings. In post-Katrina New Orleans one of the first things that the poor did to the abandoned, vacant and even to building in active reconstruction was to rip out the wiring and copper pipes to sell it as scrap.
I think you could benefit from talking to the people working these issues on the front lines.
I think you could also benefit from a closer look on the reality of the Soviet solutions to homelessness. It is not as pretty as you are imagining.