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 server on the route of it NEVER know its content, because the only mean of decrypting it resides in a memory of your client app. There is basically no way to see the contents before your client receives the message, by design. Even if that were not the case, various IM servers don’t have to send your texts to Apple, only if they want the text to be shown on a notification itself.

The only theoretical concern here may be that Apple can somehow pick the decryption key out of the app’s memory (because ios is a supervisor, obviously). But that can’t be the main reason someone swithces to android, which is a supervisor itself, and where the same issue exists.

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.

Sorry, I misunderstood that.