That would be so perverse if the government gave itself a monopoly on building housing. Just change zoning and make it legal for the private market to meet the need. Look at all the builders remedy projects in Santa Monica to see that this works.
Housing is more or less maxed out with respect to zoning laws in a lot of places. Those come from the government. Rezone to allow higher density and people will build. It is functionally illegal to build more housing.
Yes, because of NIMBYism. Existing property owners benefit from an undersupply of housing, so they lobby for restrictions in order to reduce competition which would drive down prices and rents.