I suspect many of the generalized Seattle NIMBY's champion affordable housing and decry gentrification, but when the rubber hits the road and they are asked to give up something (neighborhood charm) to actually do something about these problems (increase density to lower housing costs), they actively fight against it.
"I like affordable housing as long as I can still live in the same neighborhood and don't have to give anything up."
There is an effect where improving the consumption benefits of the city may draw new migration from other parts of the country, but it is generally accepted that this effect is much smaller than the effect of increased competition between landlords and developers.
If you think that housing somehow operates differently from a normal market - increasing supply will not change the price - it is incumbent on you to explain why you think this might be true.
As a result of denser housing, rent should go down N%, but your quality of life (if you're not fooling yourself) will also go down slightly as you have to deal with the reality of living near more people. (Longer waits at local stores, more crowded sidewalks, etc. We can pretend this is a "good" thing and that humans are social creatures, but there is a point where it becomes less tolerable to most people.)