Just want to reply to your edit. There is limited evidence that shows a direct effect on limiting supply. See the UMN study on Minneapolis Rent Stabilization.

https://www.cura.umn.edu/research/minneapolis-rent-stabilization-study

Additionally it's silly to say "price control any other product" when housing operates very differently from consumer goods. Housing is particularly unique due to it's fixed location, the lack of standardization (units can have significant differences) its status as a basic necessity, the barriers to entry for creating it, the local market effects, the cultural norms, and all of the factors around the financing of housing. The United States has had limited interventions in price ceilings that mostly relate to emergency situations, so most of what we have to go off of are functionally economic thought experiments where we set "all else equal" and even the most adherent followers of classical economics would tell you that models do not map accurately to the real world.