You seem to be conflating "rent control" and "rent stabilization", which are two different things.
Also, these economists appear to be ignoring a simple practical matter.
The idea behind most rent regulation is to avoid pricing low-income-but-necessary-to-basic-function people out of an area. These are, basically, the garbagemen and janitors and cooks and such, who can quickly find themselves priced out of anything within a reasonable distance of the city.
This results in a corresponding shortage of those services in the city, and prices for those services increase until even the wealthy -- who can still afford the higher rents -- are not willing to bear those costs to remain in the area, and begin leaving.
This can happen surprisingly quickly. In modern environments where many lease agreements come up for renewal at essentially the same time, it can happen frighteningly quickly, far more quickly than new construction can hope to "correct" the housing shortage.