CBC happened to address the issue of rent controls and housing availability today regarding Toronto (and with comparisons to other cities).
This is just a high level summary of different observations, and you can do more digging, but basically rent prices don’t seem primarily driven by supply, rent controls don’t seem to discourage new construction. Just like minimum wage doesn’t cause unemployment.
Economic models that disagree with these realities are and have always been childish, and we have the evidence to say so.