If loads of housing is built and the price of housing starts dropping, firms won't want to invest in housing because they'll lose money. Unfortunately, a lot of politically active home owners also want to keep the price of housing from dropping so this is a harder thing for local governments to put into practice.
Same reason anything else keeps getting built when the price drops. Builders are not investors. As the price drops the cost of acquiring new land to build on also drops. As long as a builder has a margin between land + construction cost, they will build.
A huge part of the cost of building a house right now is the excessive regulation. By deregulating we will be able to build houses for less money and so builders will be able to sell them for less money. In addition apartments and especially larger apartment buildings are straight up illegal in many areas even though demand is high. Legalizing these units which are much cheaper than SFHs would allow all housing prices to decrease as some of the people competing for SFHs instead buy a condo.
Why would housing be built in the first place if the price starts dropping?