If you supply housing as a government service, construction workers will play the same games as defence contractors. Do you expect rent to be cheaper?
What’s wrong with profits? They compensate for the risk and effort that comes with creating real estate. They are only too high when there is no competition.
If profits are too high, what is preventing you from creating a new house and be rewarded with those profits? Change the world so that you, and thus others, have the ability to participate. Then housing prices will be fair.