I wonder how you would register a positive case, since every public health authority has a different system for doing that part. I suppose you just have to download the app in that case?

It seems to not require an app. Apple calls this an App-less experience -- as if having an app was the still supported but the legacy way. In their description of "step 5" at the below link, users can enter an eight digit code or follow a url to inform their phone of an infection.

https://developer.apple.com/documentation/exposurenotificati...

Ah, thanks. I was under the impression that this was something that each contact tracing app set up separately, but I guess not.