There has never been a city in the US that has made housing more affordable by increasing density or legislating affordable housing. The 'why' of that is complex.
In times of massive apartment booms, rents have in fact held steady or decreased. Seattle is a recent example. There are only so many people who want to live in a particular city, so supply and demand do play a part here.
Based on your logic, we could build more apartments in a particular city than humans exist and rents would continue to go up, even if most of those apartments are empty. That clearly doesn't make sense.
Your last argument misses the point. As cities become more fashionable, not building more housing is a guarantee of increased costs due to supply and demand. In cities like SF, rents continue to rise faster than most cities. SF is also the hardest place to build anything. I wonder if those two are related.
Wow, talk about missing the point. That analysis looks at the correspondence between equilibrium compactness and price. It does not look at the marginal change in price for a marginal change in density.
I hope you can see the logical flaw in your statement "density increases housing costs".
Density and affordability are independent of each other. Affordability is about supply and demand now. Density is about supply only and says nothing about demand.
Here's (only) one example of analysis of the density<->housing affordability relationship. Spoiler alert: density always increases housing costs in the long run. https://www.newgeography.com/content/005183-what-price-urban...
There has never been a city in the US that has made housing more affordable by increasing density or legislating affordable housing. The 'why' of that is complex.