Who opens them up exactly? And that person who invests in a restaurant will leave some other job resulting in that much less product being built this something rises in price somewhere.

Rent caps just result in less people or corporations investing in rental properties. If you are alread have 90 houses and 100 people want to rent, how does caps encourage more houses?

At the end of the day the only thing that matters is how productive we are in total. Anything that is scarce will be high in price and if you mandate a lower price, then some people will simply not have shelter if you want to use that example.