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"

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.