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I found out that the Strava app is actually not well designed (and as @Jillyholli pointed out, quite useless).
In order to visualize the session you just run/rode (providing that the Strava app is correctly linked with Fitbit) you need to launch the app, wait few secs where nothing will apparently happen. But it you close the app and the reopen for a kind of magic (quote: Queen) the session will be there (at least it works 100% of times for me).
Ok, this works only if the session has been already pushed to Strava server (meaning that you have also to sync your Ionic with Fitbit app) and elaborated by Strava.
My opinion is that the app is refreshing the session list when opened (remember that Fitbit OS apps do not run in background) but fails to update the screen.