Landlords and housing crisis

In the 1970s, Britain had policies that rendered owning houses and renting them out unprofitable; private landlords sold their rental houses to cities for use as public housing. Then in the 80s the Tories specifically set out to promote private landlords again and to eliminate public housing.

It is clear how the private landlords made rental housing far more expensive. What I to not see is how this created a scarcity of housing available for rental. However, the practice of building mansions which rich people will buy only as investments has certainly taken much housing space off the rental market.