> Not via some app that is subject to all android restrictions on persistent apps, including occasional unloading from memory
Apps are subject to power and resource restrictions, but the user can easily lift those restrictions manually for certain apps, like UP Distributor.
If the app is implemented properly, it will work just as well as Google's push service.
> The OS is designed in such way requires push notifications built into its core to work properly.
Notifications are built into the OS. A network-based push notification service is not part of the OS.
It is false to say that (a particular implementation of) network-based push notifications have to be "built into the OS' core to work properly."