I'd support public housing but we don't have the tax system that existed post-war to fund it, and raising taxes enough to do so would be nearly politically impossible.

I don't think the idea that private developers are capable of lowering housing prices is actually discredited. New construction tends to be very expensive, but places that allow lots of building are overall cheaper than comparable places that don't. Upzoning slows rent growth in adjacent areas, and in some cases where it's done on a huge scale (like Sydney and Minneapolis) it actually has decreased overall rents.