New York already has had high density for a century+, but has not been adding apartments at a rate that would match population growth (and income growth) for decades.
In the Bay example, presumably if the new apartments were a one off, prices would initially come down, but would then start to rise again as more people move in and add to demand.
The solution would be to continuously add sufficient new apartments to match population growth (which could be done by liberally allowing builders to build more density to meet demand).
Which is roughly "new units decrease price if there's excess demand". Which is the classic microeconomics theory of how the price point shifts due as the increased supply sets a new equilibrium.
some pretty graphs:
https://mobile.twitter.com/georgiststeve/status/159813526427...
On that theory, New York should have insanely cheap apartments due to lack of demand.