If rent no longer rises then the profit doesn't rise commensurately with demand, so the money isn't there to build supply.
Rent control crushes supply to the degree that the rent control caps rents below the actual market rate.
If you're lucky enough to get rent controlled housing then you're in good shape. But good luck getting it when things get far enough down the timeline.
Rent fixing is the other side where supply is manipulated to extract the maximum profit. In this case the coordinated rent pricing action behaves like an oligopoly.
If rents can’t rise quickly, then the expected profit of a building goes down, which means prices for land go down, which means it is cheaper to build, which means it balances out.
If the balance is far enough off that everything turns into commercial real estate, that can be solved, too.
> prices for land go down, which means it is cheaper to build
By this logic San Francisco should be one of the cheapest places in the world to build