Hi Jamie Redacted,
I understand the confusion, especially since some Battle.net games do appear to carry achievements over to your Xbox profile.
The important distinction is that achievement syncing is implemented on a game-by-game basis. Support for it in titles such as Avowed or Sea of Thieves does not mean every Battle.net title will behave the same way. For Call of Duty, the available information does not indicate that achievements earned in the Battle.net version are automatically converted into Xbox achievements.
If you’d like this functionality added to Call of Duty, I’d recommend submitting feedback to the game publisher/developer, since cross-network achievement support is title-specific.
I hope this helps clarify why you’re seeing different behavior between these games.