> In any other market, rising housing prices would stimulate developers to build more housing
It's not clear to me that this would actually bring down prices, and existing empirical evidence of how development increases in urban cores affects prices is pretty mixed. If all else were held equal, adding supply would bring down prices. However all else tends not to be equal, and development sprees seem to often generate their own upward-spiraling demand, which sometimes actually increases prices above what they previously were, by attracting more demand than the newly added supply satisfies. Some of the concern over gentrification (though often poorly investigated) is about what those mechanisms are. One pattern is upgrading of housing stock, e.g. replacement of 100 mid-range old housing units with 200 high-end new housing units, which often has the effect of increasing all three quantities: price, supply, and demand. That can then lead to some circular migration, as wealthier people move in to the city and poorer people move to formerly-well-off suburbs which as the houses age and deteriorate, become poorer (common in "inner-ring" suburbs of American cities).
That can be good for other reasons, if you see more dense housing as a good in itself. Even if SF ends up more expensive than before, you might prefer an SF with 1 million people over one with 825k. However other people disagree that this shuffling around of rich/poor people is a net improvement. That's something of a subjective debate. However on the more objective question of whether redeveloping SF will solve the specific problem of housing costs, I'm skeptical: I suspect redevelopment of SF will not have much impact on its affordability.