I'm in the same boat: Strava is just for cheering on friends (and occasionally looking at segment PRs). My only gripe with Garmin Connect is that it unfairly negs some runs, e.g., calling my slower runs after recent snow/ice storms Unproductive.
Training Status such as "Unproductive" is calculated on your Garmin hardware device. Garmin Connect only displays what the device calculates. (I think that is a flawed approach, but Garmin is primarily a device company so that's just how it works.)
If you know that you're going to be running in challenging conditions like snow that reduce your pace then use the Trail Run activity profile and set "Record VO2 Max.” to Off. That should prevent it from changing your fitness metrics.
