> But don't push notifications basically work over a long lived connection? Of course it's better to have just one long-lived connection to Apple instead of one for every communication App, but in the end if you want real time signalling in a mobile environment you won't get around a long lived connection, don't you?

I think part of the convenience is on the server side. The server also only maintains a single connection to Google/Apple, instead of keeping millions of idle ones.