It doesn’t have to do anything, but if it doesn’t support what are considered standard features of a phone today in an open manner, it’s hard to take AOSP seriously as an open platform.
Also worth noting that when someone does build on top of AOSP, they are at the mercy of Google’s roadmap for the closed versions of Android.
The fact that this is possible is something I have commented about on this very forum for years. It is trivial for the Amazons and Huaweis of the world to build an API like this that supports Google's FCM, Amazon's ADM, etc. and provide a library for developers to use that will let them easily deploy push-enabled apps on devices across all ecosystems.
It is already usable. Nobody would claim that a Linux distro isn't open just because it doesn't have a built-in push API. You can build one on top for other apps to use, and you can do the same on Android. This is notably different from iOS.
We can have open source OSs on top of Android that build with a specific service set. Sort of how Ubuntu builds over Debian.