You're not wrong at all, but despite it all, XMPP has actually supported push notifications for years.

For various reasons it wasn't included "out of the box" in Prosody for a while and was maintained as a community plugin, but that's no longer the case with 13.0.x+ where it's bundled with Prosody and just works.

Monal is probably the most popular XMPP iOS app today.

Thanks Matt! (prodigy creator of Prosody)