The total housing supply remains static - the number of owners goes down and the tenants increase, so the S/D curve for housing stays the same. Then the wealthy consolidate the supply into smaller, more powerful groups who drive up rents via monopolist and cartel behavior (eg RealPage).
It costs money to hold on to a unit of housing. Supply is increasing (that's the premise; nobody is proposing a one-time increase in supply). How does the investor profit?