Ok, I've found it. It was StreamDeck + PilotsDeck plugin which is needed for Flight Panels' autopilot panels to work (for StreamDeck Plus). Their panels come with an installer which installs the older PilotsDeck plugin version 0.8.7 that was triggering the GSX boarding. I have manually installed the latest version 0.8.11 of the plugin and with that version GSX works as intended, no more automatic boarding + other services as soon as a flight is started. I'm no programmer but I guess the plugin was calling something that GSX listens to (something that addons with the GSX integration use).
I have tested it three times to be sure. StreamDeck client alone without the old version of PilotsDeck plugin didn't trigger any GSX action. But as soon as I started the client with the plugin installed the GSX started the boarding, calling the fuel truck, the GSX window kept popping up etc. I'm just doing my third flight with the updated plugin now and everything works absolutely fine. One happy chappy here.