We have a standardized bundle of rights that shields the person that possesses that bundle from many (but not all) price increases in a living space. It's called ownership.

If a city thinks that certain people should be protected in that way, it should buy the proprieties and give them to those favored people. Trying to unbundle ownership and tenancy to mix and match them, at a city level with only acquiescence as opposed to enthusiastic collaboration from state and federal governments, as well as quasi-governmental institutions, is not going to work well.