I generally lean towards "build more" as an answer to housing price woes. It's a nice simple story - increase supply of housing units, and prices should fall.
But it's actually too simple of a story. The character of neighborhoods and cities is affected by the number and characteristics of its inhabitants. As demand for San Francisco living from relatively rich people increases, increasingly San Francisco is filled with rich people who are driving up rents/house prices. They're also creating demand for things - shops and services - that rich people like. So a few more Whole Foods and pricey restaurants open up, making it even more desirable for rich people. And, frankly, rich people like to live near other rich people, so simply the presence of more rich people makes the area more attractive. Driving out the poors is a feature. More rich people leads to more neighborhood amenities that rich people like which boosts demand for housing by rich people. Since they're rich, their willingness/ability to pay for housing vastly exceeds that of mere mortals.
Building more in San Francisco will mean that there are more spots for people to live in San Francisco, but it doesn't necessarily mean that rents will actually fall.
This is because San Francisco is fairly small and unique. Massive infill throughout the Bay Area would surely lower rents.* But it isn't clear that San Francisco proper can realistically build its way to lower rents.
*Not clear to me the transportation network is anything close to adequate, but I don't have deep knowledge...
