> If you remove price, it will be something else besides price, such as a lottery, how politically connected you are, or something else.

The main driver of price is not "demand" imagined as how much people willing to pay for a particular house but demand as how much people are capable of paying, which is mediated by access to debt. If access to debt is clamed down on across the board, then prices will fall accross the board. People will still compete on price but as a society a smaller fraction of income will be spent on housing.