You're essentially suggesting that Android should have Tor built in. The point is that it's possible to use other push services on Android and has been done, just like it is possible to use Tor on Android. There is no reason that AOSP itself needs to implement Tor any more than Windows or AOSP needs to provide a push API. Both are possible to implement on top of the platform and have been implemented on top of the platform.
This is unlike iOS, which is the real broken platform.
Push messaging is a service, just like anonymous proxies are a service. It doesn't make sense to put them into a platform if they can be built on top of the platform. It would make sense to put something like this into iOS because it can't be built on iOS. It doesn't make sense to put it into AOSP.
Earlier you said AOSP had a framework for push notifications, a statement found to be false when checking the links you provided.
Now you say it doesn’t make even sense to put this feature in the platform, confirming that you were lying when you said it already was part of the platform.
I wouldn't be surprised if Amazon has their own internal implementation of the same thing to use in their Android apps.