All you need to understand: if rent control or price ceilings worked, why not apply it to clothes or food? I mean food should affordable to all right? This is the problem that we don't teach basic Austrian economics in high school.
And you did not address the problem. Or maybe you don't think it is a problem.
PS: I'm well versed in Austrian economics. But theory, hypotheticals and thought experiments don't represent every real, empirical issue.