The government is perfectly capable of paying for housing if it is so inclined. It is also capable of providing incentives for creating new housing in certain markets, such as discounts; the idea that this can only be incentivized by charging new and existing tenants insanely high rents is the problem. Perhaps it could get some of the money required to do this by extracting it from the tremendous number of wealthy people living in SF, many of whom are probably browsing HN right now. It would likely be easier to do that if their rents weren't so astronomical, of course...
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.
