Your paradigm is in no way connected to the reality of how people are moving. New home construction is going up like crazy in the small cities and towns people move into. To expect a small area to absorb a 50% population increase with little new construction is just not realistic.
And to expect renting to just…end? That sounds like a crazy level of privileged bubble. A huge fraction of the population is not and never will be able to afford homeownership, and expecting the government to fund their home purchases would bankrupt any nation.
That could work, but government run housing tends to become a ghetto, historically.