The pushback surely lifts the airplane. It doesn't do anything strange or out of standard: it simply sends a standard command via Simconnect to pitch the airplane up a few decimal degrees.
If it doesn't lift some 3rd party airplanes, it's possible the airplane uses some kind of custom ground handling simulation which prevents GSX to have it lifted, because the two are both acting on the airplane at the same time.
GSX has published variables that airplane developers can check, to know when GSX is pushing, to momentarily turn off any custom simulation they might have in place ( like nose wheel steering, ground handling, bypass pin, etc. ), which would cause issues while GSX pushing.
We cannot fix the problem on our side, since we cannot hack into the airplane code to turn off every non-standard simulation which might be running that would result the airplane to behave differently than a standard airplane, and it wouldn't be right if we tried that. And, no 3rd party airplane I know of has a documented way to turn off these systems remotely by other apps so, the only solution is they would do it from their own side.