The "Housing Issue" is really several issues. They all can't be solved without creating problems elsewhere.
Which problem shall we attack first?
1. Affordable housing in popular big cities?
2. Run-down public housing in popular big cities?
3. Vacant housing in areas where jobs are not plentiful?
4. Jobs which will lead to people no longer needing public housing in popular big cities?
5. Affordable mobility for those who need public assistance? (Which dictates where they eventually live / work).
I am sure there are other sub-issues.
You hit quite a few of the issues that come up with public housing - all of which need to be addressed. I put myself through college and worked afterward selling real estate in the DC area. It was quite an education. Presently I live in my original hometown, a rut belt city in a rural area with declining and aging populations. Tax base is weak, the economy is geared towards tourism - which is weak in and of itself, and an aging, retired population. For the mailman the worst day of the week is when the AARP magazine comes out - a heavy load to carry.
The experience here with public housing has been good. During the boom years when we were heavily industrialized the city built three separate public housing projects to clear slums. It worked well, of the three projects two are still open and active in excellent condition because they are publicly owned and the city maintains them with city paid staff. The funding comes from rents (33% of the household income per unit) and grants from the state and Federal government. There are no line items on the city budget for housing - it is covered off the city budget - but of course the state and Fed money comes from taxes. In addition, the third project was razed and rebuilt as a town house project - again with subsidized rents. It was a public project. We have three senior projects - apartments that are all public and operate and are funded as the others are funded. All are well maintained and fully occupied. The tenants in all the projects maintain a tenants association to keep in touch and resolve issues with project management. Lately we have also built several private projects that work because of tax breaks built into the design on the projects. Two of them are rent to own housing, that leases for seven years and then converts to a mortgage. They are six years old and seem to be working well. The other two are apartments that adjust the rent on a sliding scale for seniors. Nice buildings, privately owned, no public money to build, just tax credits for lower income residents.
As a city we have a lot of old housing that needs to be torn down - some of it is still solid but no one wants to buy it. One house is a brick and stone rambler, 2br. one bath that is being sold completely furnished - with nice furniture, no less for 25,000. USD. Yes, you can buy a habitable house here for 25K. And yet we still fill public housing.
My point is it can work well, but it takes on going work. When I saw projects in the city and suburbs of DC the biggest problem is the projects are neglected because governments do not want to put the money into upkeep, or they contracted it out to private contractors who are profit oriented but not results oriented. The biggest problem with these and other programs is that once launched, politicians stamp the problem solved and then neglect it to its death. The stories of how Trump robbed his projects by creating his own maintenance companies and subcontractors to over bill are not unique to him.
Well maintained and run we see that these projects fill a need effectively and are not breeding grounds for crime and drugs as they are often seen to be in other places. I went to school with and have known many people who were former project people who got out because they had a save place to live while their parents worked under middle income jobs or were disabled or unemployed altogether. The idea we need to eliminate is that government cannot address these issues - properly staffed and funded, it can and does.