Greece has the exact same problem. Rents are skyrocketing and people can't afford to live in their own cities. We're building more housing, but it doesn't seem to be helping.
Maybe SF doesn't want to build more housing, but this isn't universal.
It's probably not enough new housing is getting built. People don't even remember what a real large-scale housing spree looks like, because these last happened generations ago. These are transformative events. eg in Athens in the 1950s-1970s : https://www.bbc.com/culture/article/20191011-the-surprising-...
Indeed - in fact I live in a suburb of Lisbon that was just fields maybe 30 years ago - I am continually astonished at the vast scale of reasonable quality high density housing, all with schools, parks, hospitals etc - there must be hundreds of thousands of people here, a miracle of urban planning. Apparently the Lisbon middle classes fled to these new suburbs to get away from their small damp draughty downtown apartments, which ironically have now become so desirable.
My experience with most people saying "we are building more housing" is that they drastically overestimate how much housing is getting built, especially relative to yesteryear. If you look at net increase in units YoY, it's usually quite modest even for booming, "in-demand" areas.
The other part is that the same things that make it slower to build more housing also make it more expensive, all the various bureaucratic processes.
Also the kind of housing being built matters greatly. If cities continue to zone for single family homes, the increase to supply will not be significant.
Furthermore builders will need to be incentivized appropriately otherwise we'd get an abundance of luxury apartments (higher margins).
Time is money. The longer you spend time on a process, the longer you spend on people to just babysit, and the more inflation ticks up the cost of materials.
In my area of Germany new 3-4 room apartments(!) sell for 500-700k€. Waiting lists for parcels of land which must be allocated by the city stretch out for years. Old houses even in poor condition and needing major work start around 300k€.
This is not some metropolis or even medium city. Quite the opposite.
"I'd do anything for lower housing prices!"
Build more housing.
"...but I won't do that."