Hi Paul,
Taking a look at the log, there are two things going on.
The log shows you not logged into Gizmo (which handles activation of the G5). Sign in via the signup dialog. If it isn't showing, try temporarily unplugging all monitors except for your main X-Plane monitor and signing in, then reattaching them
The G5 doesn't appear to be installed for this aircraft. You need to run the G5 per-aircraft installer on each aircraft you wish to use the aircraft with. In the main X-Plane folder, there should be a "G5 Aircraft Installer" which is installed after running the main G5 installer from the X-Aviation website. Running the G5 Aircraft Installer, select the aircraft of interest (C:\newSteam\steamapps\common\X-Plane 11\Aircraft/Laminar Research/Carenado Archer_II_v3.2/, based on what I am seeing in your log file).