I feel that the GSX program needs a change to ensure that, after each ground service action, it is inactivated so that other add-ons can again become active. I am asking Umberto for this change.
There's nothing to change in GSX.
You obviously need a key to activate/deactivate GSX so, when some GSX service ended, GSX cannot give its own key back to FSX, otherwise you wouldn't be able to call it again so, it cannot be simply "deactivated" when a service is finished, because is still listening for the menu key event.
That's how any FSX addon that uses hotkeys normally works. It request some hotkeys to FSX using Simconnect, and it will constantly listen to be called in case they are pressed. It's possible, as an option, to pass the key event back to FSX when done with it, but this means if there's another addon using the same key, it would be very confusing, because BOTH events will happen so, for example, if the key used by GSX was sent back to FSX and Ezdoc (for example), wanted it to use it to change a camera view, you would see the view changing each time you activated the GSX menu, which is not very usable or desirable.
So, GSX correctly asks for a reserved key, and won't transmit the event back to FSX, and this shouldn't cause any problems, unless another addon requested the SAME key to FSX but, in this case, if they requested it in the "official" way, Simconnect would have notified to it that key is no longer available, and the other addon should handle the situation in some way.
Usually, you have ways to change the hotkey used by each addon, and this if of course allowed by GSX, so you must be sure that no key combinations are shared between addons.
Since this issue doesn't happen with ANY other addon out there, and it's seems it's only Ezdoc that seems to be affected, it seems that either it cannot deal with an unavailable key OR it uses some other method to read keys, but we cannot possibly know about it.
I can only assure you that GSX is using 100% official Simconnect-sanctioned ways to reserve and read keys AND it allows you to change its own key combination, so it doesn't need to be changed.
I saw some messages on Ezdoc forum by users that fixed this by changing the hotkey used by GSX to something entirely different not involving F12, like TAB+H.