I don't see why the ability to live in San Francisco is any different than the ability to afford a BMW or a Tesla. Should the government subsidize "below market rate" luxury cars too? What would be the incentive to get an education or work hard? Personally I feel somewhat offended when there are dozens of below market rate units available and I'm actually excluded from attempting to buy them because I earn too much money. Meanwhile there are hardly any homes available for people who can afford the houses without subsidies and we bid up the houses to obscene prices due to lack of supply.
Also, if the people who work in SF or other high income areas cannot afford to live there at current wages, there should be pressure on them to leave and work elsewhere. This would increase demand for their labor and drive prices up, all else being equal. (I know, there are other factors here, but those should be dealt with separately, not by adding more layers of complexity to the problem.)
Further, if those workers cannot afford to rent or buy, lowering the cost of renting or buying also reduces the pressure on employers to raise their wages.