To my understanding, the initial big change in Seattle was about a decade ago when they massively overhauled the city planning and zoning restrictions to effectively eliminate height and density restrictions (and NIMBY drama related thereto) in the urban core, hence the explosion in residential high-rises as well as new office skyscrapers. Recently they've been making similar but more incremental changes further out. Removing the need to build for suburban assumptions in the urban core, like mandatory parking, also made things much more sensible e.g. most people that live in the core don't drive or even need a car much, so it was silly to mandate parking that few people used. Supposedly Seattle was a bit more like San Francisco before these changes were put in place but it was before I lived here. In hindsight, these sweeping regulatory changes may turn out to be an inflection point for the city over the long term.
I've only lived in Seattle since 2010 but I don't believe the "Amazon effect" explains all of it. Most of the people, at least in tech, that are moving to Seattle recently that I've talked to seem to have two main reasons for moving to Seattle: quality of life and low (no) taxes. If Seattle can keep housing costs in check while still preserving its quality of life and tax advantages, I expect the city to do well for the foreseeable future. It may never be "cheap" but it will probably be manageable.