With Ant Agent there is an option to choose not to sync with GC. Haven't yet found that option with Garmin Express so maybe it's not there. Personally I don't care if my activities end up on GC, but I can see why it might bother some.
If you are using Garmin Express and you want the option not to sync with GC then you might have an uphill battle trying for that. With no desktop program, the only option as far as Garmin is concerned to analyse activities is with GC. With that in mind, you can understand why they would not offer an opportunity not to sync with GC.
Best thing is just t forget about it. Beat up Strava for the ability to upload directly or feed into GC like Tapariik.com