how would additional supply fix that? there is no space in the city to build more. additional supply would happen at the edges, where most people do not want to move until the area is sufficiently developed, and there are local jobs too.

a more desirable area will always be more expensive. rent control allows to keep that in check, no control will mean that even if there is enough supply elsewhere, those more desirable areas will remain expensive. rent control is needed to prevent gentrification too.