Aren't mobile push notifications an entirely different tech, one that significantly uses the capabilities of mobile carriers?

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.

SSE generally requires the page to be active in the foreground, which is not a viable solution for mobile.

Web Push is a distinct API for browsers to emulate native push notifications via service workers. https://developer.mozilla.org/en-US/docs/Web/API/Push_API

I thought it was more that the Document object had to stay alive. ...although, these days a lot of mobile browsers will kill that thing off if the page is not in the background, so yeah...

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

Yes, iOS has APNS and Android has FCM. One can use FCM for both Android and iOS as FCM is capable of abstracting away APNS. If you are building on React Native with Expo, you can also use Expo Push, which abstracts away both APNS and FCM, albeit it is not fully featured.