There are many downsides to rent control, but #2 is usually not one of them. A sensible rent control policy limits rent control to housing built before year X. This is the case for all cities in California that have rent control. For example in San Francisco the apartment needs to be built before 1979 to be under rent control.

This is a good thing for creating new housing. New apartments will not be under rent control and can command higher rent, so developers are incentivized to build new apartments.