Has anyone ever done this before? What was useful to see/experience? I'm thinking I should be focused less on tourist attractions and more on, idk, taking public transit to the library and visiting parks. Is it even possible to get a reasonable vibe in just a few days?
We actually did this, in another country. Far fewer choices, but also much less of a frame of reference to start our search with. There are many ways to approach this, but here are a few things I would encourage you to keep in mind:
1) This is for the rest of your life (at least, you haven't talked about an end date) so don't rush things.
2) With #1 in mind, I would look for a region to live in, rather than obsess on finding a city. I think regional tours are a great idea: if you like the PNW, that can be all the win you need. Get somewhere there, and then resolve to find the permanent city within the following year, taking time to see lots of places, in multiple seasons. If you get lucky and fall in love with a place, then that's a bonus, not the minimum requirement for success.
3) When you think you have a city, find a place to rent, at least at first. Neighborhoods are even more specific--again, give yourself time to explore, without the commitment that a home purchase requires.
4) On your tour, I highly suggest airbnb for stays; as you have deduced, tourist attractions are of limited value, but your neighborhood amenities (don't forget grocery options!) will affect you every day. Hotels are also limited where they are, but you can find airbnb's anywhere. And if you splurge for a whole place (apartment, small house) then you can poke around in the closets / basement / mechanical room to see what looks familiar, and features you don't even recognize, so need more investigation. Stay over weekdays, and get up for rush hour, even if you aren't regularly going to participate--see how it affects people, or if it's obvious that the infrastructure is overloaded.
It's incredibly daunting, but a wonderful opportunity. You have some good ideas to start with, and once you try the scouting trips once or twice, you will get an idea of what you are doing, and what you are looking for.