Who builds the houses then?

Who builds all those bridges, roads, power plants, and other public works projects? Maybe them?

Also government formed co-ops. Find people who want a flat and agree on rent before you build it. As the loan is paid off rent becomes substantially cheaper. The second best time to plant a tree is today.

It also doesn't need to be the entire housing stock, just enough to help keep rent low (see Vienna).

Governments directly, cooperatives or non-profit organizations. The Churches used to be pretty active in that space as well to invest tithes, with rents paying for common-welfare projects.