If high rent is the issue, vacant units should be pushed to the market instead of artificially limiting supply. Cities and residents would mutually benefit from a vacancy tax like Vancouver or similar, where international speculators are not using the units for anything useful to anyone in the city.
Two buildings are for sale. They are identical in rental income, quality, etc.. and cost the same.
The only difference between the two buildings is that one has enormous restrictions on how you own and operate the building. You as a landlord will have a cap on your potential upside in rent, but NO cap in the downside.
Which building will you buy, all other facts equal? Unless you are insane, it will be the non rent-controlled building. As such, the price of the rent-controlled building will have to be lowered to compete for a sale. Hence lower prices.
You've actually just convinced me rent control might do a city good by pushing capital investors into more useful pursuits, rather than simply "investing in land" which effectively does nothing of value for people.
I do think it's silly to treat the symptom without fixing the main problem which is zoning law and permit restrictions on building more housing stock. But why not both?
Might as well add the Land Value Tax while we're at it