Imagine two hypothetical cites, each with one million residents. The first one has half a million apartments, while the second one has ten million apartments.
Which city would have cheaper apartments?
Here's a better question: under a realistic model of building, where building slows down when landlords and property owners feel they have maximized the profit they will get out of it, by what date do you think housing prices will be low enough that current residents who rely on rent control to stay in SF will be able to live there again? 2025? 2035? 2050? What is your recommendation for those people in the meantime?
Do rich people to afford that extra housing get created out of thin air?
A) Housing becomes affordable, or B) housing stays expensive.
The latter means we added trillions of dollars to the national wealth.
By the way, please go ahead and do tax property a lot. Ideally, you'd only tax the unterlying land with a land value tax. But even a naive property tax is OK.
That way foreigners buying up local real estate for speculation means foreigners paying for local services.
As for "does not translate to affordable housing"--right now landlords and property developers have two ways of extracting money from the system: charge high rents, or pass the property on to other landlords. Maximizing profit on either of these requires rents to keep going up, and for the second there's no particular reason to actually have tenants. Beyond that, "market rate" in SF right now is nowhere close to affordable for most people and will not be for the foreseeable future, even if building greatly increases, which means that the people currently protected by rent control--those who can't afford "market rate", like the vast majority of people in the U.S.--aren't helped one bit by the creation of new housing.