Is the city "full" or something?
Putting aside the reasons (Amazon, Google, Microsoft, etc, all being willing to pay well, people wanting to live in an urban area, etc etc) the ground truth is that developers can charge a ton of money for a residence in Seattle now.
So they do.
No one's building anything affordable because hey - why not get paid a ton more for doing basically the same work by developing a luxury apartment/condo/etc.
So it's not that the city's full, it's that most people can't afford to live here anymore.
The suburbs contribute supply and absorb demand.
If developers built similar quantities of “affordable” units, we would have just as much trouble getting them, but due to waiting lists/lotteries/bribes/whatever instead of rental prices.
Given the sorry state of the apartments left after urban decay, “luxury” apartments are the only kind that won’t be a huge decrease in quality of life to middle class former homeowners.
No city views the existence of its suburbs as a serious problem. Most cities are in crisis re: gentrification and affordability in the urban core, a problem that suburbs directly solve by providing substitutes for urban apartments.