Is there a way we can avoid push notification services entirely? Can we replicate something with nostr? As much as I love Apple and Google, I want to be as independent as possible.
Discussion
Pokey avoids it. But Apple won't allow you to go around itself.
Hmmm... Despite all of Apple’s walled garden limitations, there are unofficial Syncthing clients for iOS, and, somehow, they were approved for Apple's App Store.
I think it’s mostly a matter of persistence, lucky and endless patience when dealing with Apple’s “reviewers”. But if folks are able to get a encrypted two-way file sync solution running in the background aproved by Apple, there has to be a way to get something like Pokey approved on iOS.

I didnt know that. Bad example from a technical feasibility PoV then. The point about approvals is still true though.
Just looked at their FAQ. Yeah, whatever they are doing works well for eventually consistent sync, not for potentially time critical notifications:
Background sync
Apple iOS restricts apps from running continuously in the background, but apps can run for short times sporadically. Möbius Sync uses various methods to invoke background behaviour. The minimum interval between quick syncs and power syncs can be configured under Settings, but iOS schedules background activity in an adaptive manner that is not predicatable and sometimes counter-intuitive. It may take 24 hours to start to sync but you can expect a total of 1-2h of sync activity per day once stable.