Unfortunately SF doesn't have a good track record in this regard judging by the Midtown Park Apartment fiasco: https://www.sfchronicle.com/bayarea/article/Residents-of-S-F...
I too think a Singapore-style development model would be worthwhile trying. That is, subsidizing development of housing that would eventually be sold and transferred to tenants, with or without restrictions such as market value percentage caps on resale, stricter rent controls if leased out, etc. It's almost the only viable way forward because there's simply not enough public funds, now or ever, to make a dent in the supply problem without leveraging private financing. That's the beauty of the Singapore model.
But for various practical (e.g. management expertise) and legal (difficult to constitutionally restrict the benefit to pre-existing city residents only) problems, to do this at scale would probably require the joint efforts of the Federal government, state, and city. Or perhaps some billionaires would be willing to step forth to try the experiment themselves, rolling property sale revenues into new projects.
Ultimately, though, this doesn't solve the NIMBY problem. And if you can solve that problem, there's less need and pressure for attempting big experiments like this.