I feel that you missed one basic aspect of economics. Competition is one reason prices might go up. There are other reasons, which are relevant here. Monopolies, collusion, price fixing, goods that people can't live without, speculation, those are also reasons that prices go up.
In the housing market, it's not fair, it's not free, this isn't a basic supply and demand situation.
I was having trouble understanding what you meant because you didn't think about the obvious implications of millions of properties being unloaded in a short time.
If the number of landlords drastically increases, which would happen when you have mass property sales, then there's more competition, and rent goes down.
Or, depending on your setup, the government seizes some of the properties that people refuse to sell, and turns them into public housing. This also drives rent down.
So then, what happens? Oh yeah, both buyers and renters win. Was that clear enough? Perhaps I should write in all caps.