to clarify, once you set up Syncthing, it runs in the background even in gaming mode! You do need desktop mode to add new games to syncthing, but I think there's not really a good way around that since every non-steam game has its own save file location. Plus you need desktop mode to install non-steam games anyway.
The one downside to syncthing is that it needs two machines running at the same time to sync them, my workaround for that is to have a raspberry pi running it at home all the time :p