@nassoscal I would risk stating what you may consider obvious: the proper way of recording activities, so that they show in Strava is:
enable export of activities from Suunto App (SA) to Strava (in SA)
sync activity from watch to SA
SA sends the activity to Strava )this should happen automatically)
Strava (app or web) should not need connecting (syncing) the watch to the app.
Please note that occasionally, the sync process (from SA to Strava) takes a while (anything from a few seconds to a few hours).
Sorry for talking the basics, but your original post wasn’t overly clear.