Is anyone arguing that price drops "lead to" (i.e. happen before) increases in supply? It seems to me that it's the other way around: supply increases "lead to" price drops. This is what supply advocates hope to see in HCOL cities with expensive housing.
The author is concerned that builders will stop building when prices fall, but that's rational. Suppliers should respond to changes in prices.
Other policy goals are possible. For example, instead of "affordable housing", your goal might be to help the home building and financing industries never experience a down year.