I don't know about the implementation details, but even the use case is slightly different, as the discussed techs are also meant to transfer larger amounts of data. At the same time, mobile push is more about triggering synchronization than transferring large amounts of data.

However, since the discussed techs all have major problems with mobile connections, I still think it should have been discussed in more depth.

You can do mobile push notifications to a phone even if they aren't connected to a mobile carrier.

That said, I'd argue SSE is the browser equivalent of mobile push notifications.

iOS and android push does not rely on mobile carrier capability afaik, it has its own service