> enough to fix the problem
I think we need to define the problem before we start arguing solutions. In my view "can't afford the rent" is a poverty problem, not a rent prices problem. So, I want anti-poverty solutions, not rent price adjustment solutions. I view "the rent is too high" as a distinct problem from "can't afford the rent." So in my mind, rent price adjustments via price regulation are inappropriate for both problems.
> so that only productive activities ... are profitable
I understand taxing negative externalities, but this sounds anti-democratic. Who decides what's productive? Let's put an appropriate tax on pollution (broadly defined) and let the market figure out what's most productive. That's what it's good at.
> why oppose them?
I don't know of any instances where rent control didn't affect mobility. Are you considering the effect on immigration and wages in other cities? If they didn't affect mobility, then almost by definition they had no effect whatsoever. They'd have to be covering such a tiny portion of the market that they can essentially be ignored, both cost and benefit. Which makes them a waste of bureaucracy.