You know, completely semantics here, but a corrupt corporate oligopoly housing market isn't fundamentally different than a corrupt centralized government run housing market.
Both problems in both systems are fixed with legislative action, in one case rent control and in the other changing out officials and restructuring offices.
Here is a practical example: In China, everyone gets a home. Except there are hundreds of millions of homeless. Because they don't get to pick where the home is, and for a lot of them it is not in the city they work in.