Rental payments are higher than mortgages in most large cities. And those renters usually can't even qualify for a mortgage, despite paying more in rent.
* any and every landlord immediately pays taxes based on the asking rent, no matter what rent it is.
If a landlord is asking $2500/mo for something, the government immediately values the property based on whatever the formula is based off the asking rent, even if the landlord cannot fill the space.
To work to decrease or cap rents this would have to be a dramatically increasing progressive tax. It would have to be a complicated progressive tax to work with both small, less desirable residences and mansions.