Private equity goes to housing because there is a return there.
A housing can either be an investment or be affordable. It cannot be both. Those two things are completely at odds.
When something is an investment, it must become unaffordable over time. There is no way to square that circle! Unfortunately, housing-as-investment is more or less baked into our cultural DNA. We do not need to unwind the supply problem; or the government regulation problem; or the corporate problem. We need to unwind the idea that a house is an investment.
Good luck with that. It's the air we breathe here. There's nowhere to go but up.