Indeed they would, however, the underlying asset value would drop to allow for market rents. That's the point.

In a free market, yes. But not when the demand is so artificially constrained. Rents go up and people are forced to either pay a higher percentage of their income to rent, or move farther away.

Housing is not a free market by any stretch of the imagination, so if you just move one lever you don't always get the response you would in a true free market.