Why let private developers build at all? The market provides perverse incentive to not house everyone.
How so?
Because central planners do not have the information processing feedback loops necessary to distribute all of the complex network of component commodities.
Truly there is some scarcity of resources in any given place that takes energy to extract, transform, and transport to the desired location.
Because everything cannot be everywhere, there will be unmet needs in a society.
The "free" market economy allows for a natural price discovery of everything which is more likely to not have shortages or overages in any given spot.
Granted this is pure basic economics. It does not imply when for a toy example, there are 10 people thirsty and a baby and there is a scarcity of milk, 1 cup, the milk should go to the highest bidder. obviously it should go to the baby.
Same thing with housing. There is a scarcity of all of the components that go into housing and administering the amenities that houses need in a city. Central planners do not have a good track record but with careful tracking and modulating feedback loops we might be able to meet more people's needs that an otherwise regulatory captured monopoly of vertically integrated home builders so that everyone has a better chance at a home.
But again not everyone can live everywhere logistically so how do you fairly distribute it? Obviously hunger gaming it in a cut throat capitalist world isn't working out too well but we do have the largest number of people ever on the planet so its a pretty decent system so far
To create artificial scarcity on a free market, it should be either a direct monopoly, or a giant cartel that includes every single developer and has mechanisms of punishing the outliers.