Corollary: notifications should be a public service whose only purpose would be to maintain that TCP connection and have a server as a broker, with clear control given to the user over which app can use said service and be allowed to wakeup their device.
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.
