Basically, the state would buy the houses and rent them out.

We used to have this (local governments building houses and renting them out), privatisation forced these organizations to only cater to the lowest incomes. Which honestsly is really great for them, the problem i described before is mainly one of the younger middle-class (finished education but not yet in a position to buy a home). Iirc Denmark and Austria have social housing for a much larger group. But it does seem to come down to money: is the upperclass willing to subsidise housing of the classes below them.