Removing or abstaining from rent control wouldn't address the fundamental problems. Indeed, I'd argue that expanded rent control could be used as a stick to pull the renters lobby away from NIMBY activists who have convinced renters that any new construction is against their interests.
Rent control can be viewed as a type of wealth transfer (social welfare) that all economists implicitly presume will be used to ameliorate distribution and dislocation inequities. Because "tax" is a four letter word in our society and cash transfers both stigmatized and disfavored, we rely on regulatory machinations like rent control or trade tariffs to implement the kind of wealth transfers that economists all presume would be done with taxes and cash payments.