Adding more housing will lower prices, and lower prices will increase quantity demanded. Yes, more people will be able to stay in and move to the area. That's the point. You've not made an argument for why additional housing would increase demand (i.e. desire to move, willingness to pay).
Road expansion boosts capacity just fine. Not as cost-effectively as public transit, but it works. What it doesn't do is fix congestion. I'd argue that there really isn't anything like congestion in the housing market. Homes are discrete; you have one or you don't. There's no such thing as "extra" people in the neighborhood shrinking all the existing homes.
Belatedly... yes, you are right, I haven't described any shift in the demand curve. But so what? I was responding to a suggestion that the reason there is a housing "issue" is that demand is relatively inelastic, and exceeds supply, which is also relatively inelastic. My point is simply that demand could actually be fairly elastic, and so we should not describe the problem in this simplified way, because if that's the case, increasing supply will not reduce price very much, and so increased housing won't solve many of the problems the YIMBY cause is hoping to address which are caused by high housing prices.
In the road analogy, congestion on roads is akin to a price--people are paying for trips with their time. I was trying to make the point that adding more roads doesn't bring down the "price" of a trip substantially, just like adding some more housing in SF may not bring down the price of housing substantially.
OP responded by noting that, in their view, having more people in SF is a good thing in and of itself. If you agree, then we have no argument (this is almost surely a result of building more housing). But if you think that adding more housing would be good because it would allow low-paid workers to live closer to where they work, or many of the other reasons advanced in this thread, my point is simply that the analysis of this policy proposal requires more nuance than the kind of N > M analysis OP was conducting.
Road expansion boosts capacity just fine. Not as cost-effectively as public transit, but it works. What it doesn't do is fix congestion. I'd argue that there really isn't anything like congestion in the housing market. Homes are discrete; you have one or you don't. There's no such thing as "extra" people in the neighborhood shrinking all the existing homes.