I'm just seeking proof of that. There are limitless alternative even contradictory theories. Why, other than matching preconceived world views, do we choose one over the other without evidence? In the Cambridge example 25 years have passed but there is, as you say, no additional housing and prices have risen further. Nor was there any burst of economic growth. In what way and to what end has the market become healthier?
And if it does not increase supply of new construction then it is zero sum gain. It merely benefits one group, in this case landlords and new residents, over old residents and tenants generally. And the harm done is not temporary, the increased outlay in housing is permanent regardless of who is paying it. And wealth is transferred to unproductive literal rent seeking from productive activities such as consumption, employment and businesses (who now have bigger payrolls just to pass through employees to housing).
And now, with transient population replacing a permanent one, interest in long term undertakings such as mass transit and community undertakings such as environment, homelessness or even economic stimulus also wane.
The correct solution is to add more housing which evidently is unconnected to rent control. Certainly on existing units.