You don't even need public housing for this. The higher rents are the more profitable it is to construct new housing. The issue is currently that we literally prohibit it with zoning rules, then make construction less profitable (and therefore less common) by using property tax as opposed to LVT, then (to a much lesser but still relevant level) make construction more expensive through various regulations and trade licensing rules that increase construction costs.

Make it cheaper and easier to build new housing and you'll see a lot more new housing.