So almost literally how iOS and Android notifications work except I guess jointly owned and operated by by Google and Apple? Because public services like this are necessarily funded and operated by the companies that have a financial interest in them existing.

Not really: you can't use the service alone, without any other software. The software is not open source. And finally, the user has no control over which apps/services can send them a notification (or really, a device wakeup) at a given moment.

Those are downsides in many situations but the existing systems match your original description perfectly, unless "public" has a special description you didn't communicate.