The wikipedia entry (and the official entity?) is closer to a US state or large county, with a size of 31,000 square miles (this would put it at ~the 40th largest state).
If you take the urban population listed there and divide by the listed urban area you get 4,250. A bit above 960.
What I see people missing here is the transportation aspect - land is only scarce in San Francisco if it’s permanently limited to a 7 by 7 mile square. Transportation increases the viable size of where you can live to access the city. So make land less scarce by building fast transit from other places.
That all said, it’s also a joke to think that SF uses its land wisely. What’s at the 24th st mission BART stop? A 1 story McDonald’s, what a great use of prime real estate to protect a treasured cultural relic.
Of course it's cheaper to house people when land isn't scarce.