AOSP is not Android. What Google distributes to OEMs and its direct customers is not AOSP. It is AOSP and the different Play Services including the Play Store and the Google applications with contracts mandating that they have to be provided to customers.

If you think people talk about AOSP when they talk about Android, you are always going to miss the point of these conversations.

The issue is the locked-in requirement to rely on the sole source of push notifications built into the system. Running your own persistent app is an ugly workaround that doesn't work more often than does.

Also, such apps are unloaded from time to time even on stock non-modified Androids.

AOSP is horribly crippled by Google, as they move more and more features towards play services. Location, push notifications, etc.