Developer-owners are the exception. Most develop to sell immediately, or after 5-15 years. Their selling price is largely dictated by the cost of debt, land, labor, and materials, and projected NOI. Then of course, there are developers fees and market fluctuations. Of those factors, only NOI has any direct connection with rent control.

If rent control doesn't take effect until year 15-20, allows for inflationary increases + an additional x% buffer + and generally includes vacancy decontrol, which is basically every rent control/stabilization plan in the country, then there is functionally no connection with profitability to develop. Rents are unrestricted for years and minimally constrained with opportunities to increase infinitely in the future.

Rent control does have a strong connection with is developer/landlord fear mongering. What arguments like yours do is take developers at their own word rather than recognizing that they are inherently regulation averse entities that will always argue that anything that stops them from making the maximum possible profit isn't just bad for business, it's bad for society. There's some nuggets of truth there among the lies and as we do primarily rely on private market rate developers and landlords to produce and sell us housing, incentivising their profit motives can be useful. But we don't have to be rubes about it or let them walk all over us.

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.