However, just at the same moment the aircraft doors start opening, I get the same message like you would get when overstressing the aircraft in flight as seen below.
Clearly not caused by GSX.
At NO TIME GSX ever changes, affect or touches the airplane status (any of its variables) except in ONE single moment: when you see the message about the Bypass Pin being connected. At that time, GSX freezes the airplane and takes control of it for the whole Pushback, and will release it when you'll see the message about the Bypass pin being attached.
This is the only time where GSX could theoretically affect the airplane. Anything out of the ordinary happening in other situations, is not caused by GSX.
If this doesn't happen without GSX, it doesn't mean it's caused by GSX either. It might be some kind of custom code running in the airplane that is doing something related to its own internal GSX automation, and it's not even granted it's completely caused by the airplane, some other add-on might interfere, like FS Realistic with the "Ambient Wind" feature, which should always be disabled.