> we should be building more of it to fill demand, and in turn bring prices down.

That's not really how construction works in the US. All construction generates higher prices than existed before (this is explicit intentional policy, at both the local, state, and federal level).

There is no such thing as new housing in the US that lowers property values, therefore there is no such thing as a new building that lowers "prices". No one does this (except perhaps, non-profits and such). And even if you wanted to anyway, no bank would ever finance it.