They do not have enough capital to do that. Housing is such as large market that eventually anyone but the state would run out of capital. And it is not actually even that profitable... There is much better investments.

You don't have to buy out the whole market. You buy the houses that are likely to turn over, and then you make sure that they don't until you want them to. Housing and rent is set at the margins. If the average price in your area over the last year for comparable housing was $300k, and the last three comps sold for $400k, your asking is $400k.

It's exactly the situation you'd expect with record high prices and low sales volume, which is where we're at.