People are able to move around, to some degree, so housing prices are a function of supply across most of the nation. Or at least the desirable portions.

Rent control on the other hand has mostly local effects.

Which means, rent control can push prices down and keep them down. There is indeed a supply reduction, and prices on average will go up—but not in the rent controlled area.

It’s still a poor idea, but it requires centralised planning to avoid.