Sort of. There's nothing stopping you from using script inside a Service Worker to send those notifications. Calling them push notification is troublesome as the Web Push API and Notification API are actually two entirely separate things.
Yikes, that really seems like abuse of HTTP/2 semantics to me. My understanding has always been that server push in HTTP/2 is only intended to prime the client cache for requests you know they're about to (or likely to) make. [1]
> Pushed responses are always associated with an explicit request from the client.
I'd rather see additional frame types defined that allow for bidirectional communication (i.e. websockets 2.0) than overriding and conflating the PUSH_PROMISE frame type.
I've built a plug & play app for Web Push Notifications. It comes with an analytics dashboard and an interface for users to manage their subscription. If anyone is interested to try it, you can see the demo at https://www.pushroll.com