It might be unsatisfyingly simplistic, but if prices are high in an area because a lot of people want to live there, building more housing seems like a fundamentally better approach than demanding everyone who wants to live in San Francisco instead accept living in Mississippi. Even if prices stabilize back to where they were before, it would still be a net win that more people than before get to live in a desirable place for the same cost. I'm genuinely not "sneering" at Mississippi here. But if you want to live in San Francisco, Mississippi is probably not going to be a reasonable substitute and maybe making it easier for you to live in San Francisco is a reasonable policy goal.

But history and lots of examples suggests prices are unlikely to remain the same if you build more supply, which makes sense from an economics perspective. There aren't an unlimited number of people wanting to live in a particular area at a given price point, otherwise they would likely have driven the price higher in the first place. Induced demand is a problem for highways because the cost of driving that particular route at a particular time is essentially zero, which his not really true of moving to a different area.

Isn't SF infamously a SFH zone that makes no sense? If you razed the whole city and rebuilt it like Manhattan, you wouldn't have 25% more inhabitants, you'd have 25000% more. And if you improve public transit (I heard BART is some that's pretty good by American standards but not by European standards?) you can drastically widen the area that feels like the core city (just take a look at Berlin which I'm familiar with) giving maybe another 1000% multiplicative increase.

Every time I'm in Berlin I love the public transit system there. I think that's because they have a relatively dense network of lines through the city center, and the trains run every 4-5 minutes which means you don't have to check the timetable before going to the station, you can just show up. I haven't visited London or NYC though. I know some cities with more modern train signaling have trains every 1-2 minutes so there is basically always a train at every metro station.