Max,

Edge has supported Web Manifest for a while now. Any site served via HTTPS with a valid web manifest file is eligible to be in the Store and have access to the full platform APIs. They are called Hosted Web Apps. Netflix & Spotify Windows Apps are HWAs. They should only get better when SW ships.
And I agree they need to ship SW very soon. They have been teasing for way too long. Hopefully at the Edge Summit in September (fingers crossed)
Part of their problem, IMO is they are under the platform (OS) division and not the DevDiv (developer). So they do not have the freedom to ship out of band. But what do I know ….