Allowing construction will alleviate pressure on the system as a whole, but, perhaps due to a failure of my imagination, I'm not seeing ways in which it won't cost something. I think it's worth it, mind you, I'm just not seeing how to make it $0.
In 2020 SF, even the upper middle class struggle to pay for housing. If you, as a first step, allow the housing stock to double, twice as many people can afford to live in the city, and life in SF becomes much more accessible.
Sure, there will always be people out of luck needing some assistance. But it should dwindle down to a smaller core when there are places to live.
Personally, I don't believe in government run housing. It's better to help people with rent money etc. But of course, none of this will ever happen in SF, so my opinions don't matter.