iOS or Android push notifications (can) use SMS for notifying the client that a new message is available ? That’s lovely. Do you have any links or any keywords to find more ? All I can find online is that iOS uses TCP (XMPP in fact :o TIL. )
(Said differently, the radio firmware got good enough that it doesn't need a special "wake up" packet anymore, it can recognize a packet to itself in a low power state.)