Yes build more. Also regulate the oligarchs until they no longer exist.
Me: "should we stop allowing private equity to hold property?"
My Strawman: "nah let's build more houses to be snatched up by private equity, that way eventually maybe they'll be really nice and decide to lower prices for us someday due to the infallible balance of supply and demand that definitely hasn't been rigged to hell over the past half century. They certainly won't just continue to rig the system to benefit themselves"
Housing demand will always be > 0 as long as population is growing and hence there can never be a oversupply.
How has "supply and demand" been "rigged"? A lack of supply isn't evidence of "supply and demand" being "rigged", it's it working exactly as predicted, not any different than oil prices going up when there's some geopolitical event threatening supply.