IMHO UnifiedPush is just a poor re-implementation of WebPush which is an open and distributed standard that supports (and in the browser requires, so support is universal) E2EE.
UnifiedPush would be better as a framework for WebPush providers and a client API. But use the same protocol and backends as WebPush (as how to get a WebPush endpoint is defined as a JS API in browsers, would would need to be adapted).
A push protocol. You get a URL and post a message to it. That message is E2EE and gets delivered to the application.
A way to acquire that URL.
A way to respond to those notifications.
My point is that 1 is the core and already available across devices including over Google's push notification system and making custom push servers is very easy. It would make sense to keep that interface, but provide alternatives to 2 and 3. This way browsers can use the JS API for 2 and 3, but other apps can use a different API. The push server and the app server can remain identical across browsers, apps and anything else. This provides compatibility with the currently reigning system, the ability to provide tiny shims for people who don't want to self host and still maintains the option to fully self host as desired.