We could debate how this works, but since neither of us is an economist, that would be as silly as a couple of non-scientists arguing about climate change -- when the overwhelming majority of scientists are all saying the same thing, just listen to the scientists.
In this case, an overwhelming majority of economists are saying that increasing the supply of housing in our cities would bring prices down.
Even if they're all wrong, and we increase the supply of housing by 100,000 units and prices stay the same, that's 100,000 fewer families that get displaced.
It does mention that it has a strong elasticity for demand. Particularly higher for medium+ income. Since its very hard to measure future elasticity of demand (i.e. SF now known to be expensive, if housing was built agressively and it became "famously cheaper" demand could be even higher) its possible that prices wouldnt go down much.
The correct answer to the OP's concerns is that the goal is not to lower rents, but to increase the total utility: you do that by providing more and better housing. New buildings at same rate for the same amount of people is still better. And naturally, same rate and more people is also better.
There's no doubt that San Francisco needs a massive shock to the supply curve.
In this case, an overwhelming majority of economists are saying that increasing the supply of housing in our cities would bring prices down.
SF's Chief Economist: http://www.sfhac.org/wp-content/uploads/2014/11/Ted-Egan-Pre...
California's legislative analyst's office: http://www.lao.ca.gov/reports/2015/finance/housing-costs/hou...
The Obama White House: https://www.whitehouse.gov/sites/whitehouse.gov/files/images...
Even if they're all wrong, and we increase the supply of housing by 100,000 units and prices stay the same, that's 100,000 fewer families that get displaced.