In my experience the design of a website is often done without a browser in mind at all. A designer uses an application like Sketch or Photoshop to mock up what the site should look like, and once the client has signed off a web designer/developer figures out how to build it in a way that works in all the target browsers. Starting with what the browser can do sounds like a much better approach, but that's going to take a paradigm shift in what both designers do and what clients expect. I can't help but think that's going to limit adoption of 'new browser first' for a long while.