Earlier quoted context omitted.
Sure, but there still needs to be a server somewhere that has the app developer's client certificate, connected to Apple, that knows when you get a DM and can send that zero-content "wake up and check your DMs" packet to your phone via APNS, which means it needs (given current apps permissions models) access to your DMs without you.
There is no need for this “server somewhere”, and APNS can’t have your message, mainly for two reasons: First, that “server” is Signal server itself. When a message arrives at it, it simply commands Apple to wake up your app, for it to call home, and that’s it. It is preconfigured to do that (and GCM as well). Second, all Signal messages are end to end encrypted and neither of Apple, Signal, Google, your ISP, any ser…
In a centralized system where the mobile app is made by the same people as the IM server, of course the IM server can send a notification to the app (such as is done in e2e stuff like Signal, or in non-e2e such as Telegram).
That falls apart when you're talking about, say, ActivityPub on an instance you run yourself, or one not operated by the vendor of the IM app. The developer's server can't know when to send you a notification without having some knowledge, from your own e.g. IRC, ActivityPub, or Matrix server, that you have received a new DM.
The Matrix/Element people have addressed this by running one centralized notification service for every single client of the Matrix/Element iOS app.