More housing is very likely to keep rent steady in san francisco. And you can also have increased housing supply, increasing rent prices and decreasing housing quality.
The core issue of Nimby-ism is the tax system. If you make more housing, the rent-value of houses goes down, but the rent-value of land can and will most likely shoot up. Land that can be constructed on increases in value, and if you keep taxes fix, it means more workers and more renters pay more taxes (from their jobs, from their businesses, from their consumpition) that goes into the pockets of landlords in the form of public services.
SF city spends 9 billion dollars a year. Policemen make 400k, firemen make 200k+, Bart employees make 300~400k. And you have landowners that bought property 30 years ago, and make 60~70k a year, pay little to no state and federal taxes, and get all the profit from the insanely inefficient spending.
SF and California is doomed for deeper reasons than Nimbysm: It is mainly saved by migrant labor and good weather, but eventually that will not suffice.