Yes you interpreted nsec.app's architecture right, it depends on a server that detects unanswered requests and sends push notifications to wake up. iOS is actually broken with a PWA, the issue is that PWAs don't receive push messages on time after the last iOS update, which makes nsec.app basically unable to work in the background.

Offline signer like Amber using native inter-app comms is obviously a much better approach, but it doesn't work on iOS or across devices, so if you're using desktop you'll still need something like nsec.app. Confirming every action is a UX nightmare for active daily use - people complain very hard here every time Alby popup shows up without much necessity. I think it's inevitable for iOS to come up with inter-app comms API someday, so maybe all we need is just wait a bit.

It's great that you're exploring options on iOS and I bet there will be useful building blocks created as a result, so please keep doing it and sharing your learnings!