Earlier quoted context omitted.
Yes. And also no. Today, if I cannot pay, I get no housing. Taxes allow us, as a community, to decide that we should all chip in a little bit extra (commensurate to our means) to make sure everyone has a roof over their heads. Also, government operated facilities do not have a profit motive, so there's less incentive to maximize the cost of housing.
Hilarious how your system gets more and more complex requiring more and more rules. You’re basically taking a market that has an artificially constrained supply and trying to replace it with an entirely planned system. Why?
And to be clear, I've proposed a system of adjusted incentives followed by but not necessarily required to be followed by a more planned system.