I don't doubt that demand exceeds supply. San Francisco also is not really a representative sample of US (or global) home ownership vs rental and so on.
All that aside, I don't really see the line of argument that links my proposed idea to "more restrictions". My initial thinking was something like (basically): current market forces (especially in SF) favor the consolidation of ownership in the hands of the few people who can afford the large capital investment required to build new housing.
IOW: home prices are ridiculously high at least in part because the future value of rent extraction makes these homes good long-term investments for people who don't live there. See also: removing units from rental for AirBnB service in the long-term (sources differ on how many units in SF are lost to permanent AirBnB hotel status, but they seem to all agree that at least some are off the market for this reason).
So, what happens if this element of the market is removed? I know, I know: it's central to capitalism, I'm becoming a communist, etc etc. What happens when the supply - demand relationship is between residents and other residents? Do home prices trend upwards? Does the current bloc of owners and rent-controlled residents stop acting to fight against further expansion?