It has, but is almost undoubtedly slowing as Seattle becomes more expensive and housing becomes more scarce (in other words, as it becomes New York-like). Massive inbound migration was possible because pre-Amazon, housing in Seattle was comparatively plentiful and cheap.
Seattle has a housing crisis because it has large amounts of the city with extremely restrictive residential zoning and refuses to compromise. They have large areas where multi-tenancy housing can't exists. Additionally, in the US everyone believes housing is not a depreciating asset. There is a reason Tokyo is the largest city in the world while being significantly cheaper to live in then similar cities.
Of course. This is true of New York as well, especially outside of Manhattan. I, too, would prefer a regulatory environment that allows for denser construction. None of that changes the fact that the regulatory environment is what it is, and as a practical matter it limits the extent to which people are able to move to either city to work for Amazon.