This is an ongoing argument in Northern Virginia (which is not quite as expensive as SF / Seattle / NYC, but probably only one cost tier below that) and the major arguments are basically the same issues that apply to all "just build more housing, stupid" proposals.
Basically, if you suddenly build a lot more housing, you'd start to strain the infrastructure of the community in other ways. That strain is really, really unpleasant to other people who share the infrastructure, and so current residents -- who are often already feeling like things are strained and getting worse over time -- would rather avoid making things worse. The easiest way to avoid making things worse is just to control the number of residents, and the easiest way to do that is to control the amount of housing. If you don't live here, you're probably not using the infrastructure. QED.
In many ways, building more housing is the easiest problem to solve when it comes to urban infrastructure. Providing a heated place out of the rain just isn't that hard, compared to (say) transportation or schools or figuring out sustainable economic balance.
Existing residents are probably (very reasonably) suspicious that once a bunch of tiny apartments are air-dropped in, and then a bunch of people move in to fill them up, that there won't be any solution to any of the knock-on problems that will inevitably result -- parking, traffic, school overcrowding, tax-base changes, stress to physical infrastructure like gas/water/sewer/electric systems -- until those systems become untenably broken. I mean, I can't speak to Seattle, but those things are already an increasingly-severe problem today, with the current number of residents, in my area, and people don't have much faith in government's ability to fix them; so the idea that the situation will be improved once everyone installs a couple of backyard apartments is ridiculous. (And then there are questions like: how are these backyard apartments going to be taxed? Are people who move in really going to pay more in taxes than they consume in services and infrastructure impact, or is this going to externalize costs via taxes on everyone else? There's no clear answer to these questions, and people are reluctant to become the test case.)
If you want more housing, you need more infrastructure. If you want more infrastructure, either you need a very different funding model or you need better government and more trust in that government. Our government is largely (perceived to be) broken, and public infrastructure is (perceived to be) broken or breaking, and so the unsurprising result is that nobody wants to build more housing and add more strain to a system that's well beyond its design capacity anyway.