My guess is that it has to do with that private relay because OAuth isn't too complex by itself. During the OAuth flow they probably collect the user preference, (if needed) go out to the relay service and get a generated email, and POST back to their own service with the preferred email to use in the token.
If that's it, it's about as bad as doing password authentication in JavaScript and passing authenticated=true as a request parameter.
Edit: Looking at the OAuth picture in the article, my guess would be like adding a step in between 1 and 2 where the server says "what email address do you want here" and the (client on the) user side is responsible for interacting with the email relay service and posting back with a preferred email address. Or the server does it but POSTS back to the same endpoint which means the user could just include whatever they want right from the start.
The only thing that makes me think I might not be right is that doing it like that is just way too dumb.
AND I'm guessing a bunch of Apple services probably use OAuth amongst themselves, so this might be the worst authentication bug of the decade. The $100k is a nice payday for the researcher, but I bet the scope of the damage that could have been done was MASSIVE.
Edit 2: I still don't understand why the token wouldn't mainly be linked to a subject that's a user id. Isn't 'sub' the main identifier in a JWT? Maybe it's just been too long and I don't remember right.