Price controls dont work. This is economics 101 and tested many times in the past and as of recent. All you're doing with artificially constraining price and other regulations is reducing supply in the long run. The issue bubbles and then eventually bursts.

However, the issue in congested areas is a matter of physics and people unwilling to relocate. Its not from renters simply fulfilling the pricing demand. I mean what exactly is the long term play in areas like this? Just keep stacking people on top of each other into infinity? Its obviously not realistic.