There would be high demand for buying land to build new homes on. Or buying land with old low-rise buildings, and replacing the buildings with higher and/or more efficient design, to provide more homes on the same piece of land.

But even if the owner would like to sell the land, and buyer would like to pay a lot of money, this is usually not allowed, for political reasons.

If let to operate more freely, free market economy would be efficient in increasing the density of homes in high-demand areas. This is very much not happening in San Francisco.

If people are willing to pay Manhattan-prices (and even more), the free market response would be to build at Manhattan-density.