Regarding the .25m walk at the airport, for those of you who are mobility impaired, or just plain tired, the POS (no, not that POS) operates an electric buggy from the skybridge to the terminal to the light rail station. If you find yourself on the Link around morning or afternoon rush, you will get a seat out of the airport, but by the time you get downtown you will have a very packed rail car.
The first group I’d honestly say link to monorail and walk to the hotel. You don’t even need to leave Westlake station to catch the elevator to the monorail, use the savings on a better dinner or dinks that night. The Hyatt House is in a safe area, but there’s not a lot of dining options, if you’re up for it, cross through the Seattle Center to Lower Queen Anne (also known as Uptown, but nobody in Seattle calls it that) where there’s a plethora of restaurants. I hope this helps.