What about segregation? In Sweden most people are in favor of rent control. They don't want the rich being the only ones who can afford living in nice areas and let the poor be lumped together in bad areas.
This also gives better public control of who receives the subsidised housing, usually a combination of needs assessment to qualify and then a queue system.
The important thing we have learned from the 70s/80s (different timing in different countries) is to spread the public housing throughout the community, rather than concentrate it into a single location. That is, rather than have a large apartment building that is purely public housing, have a requirement that a certain percentage of properties of every new development will be made available as government subsidised public housing.