Airplanes generally do not cruise by compass, but using an inertial system (INS). Inertial systems (gyroscopes) have the advantage that are less sensitive to magnetic anomalies, compared to compasses. They also indicate geographic north, and not magnetic north, which is less useful.
Because airplanes fly on a round and rotating Earth, the gyros of the INS need to be realigned periodically. With older airplanes, the pilot needed to intervene every -say- 15 minutes to realign the gyro manually. With modern airplanes, this realignment is performed automatically.
In a flat and non rotating Earth, the gyro of your airplane would not change direction and would not need to be realigned. It would continue to track the original course.
In a round and rotating Earth, the gyro shows two types of movement: the drift rate and the transport wander.
The drift rate is the effect of Earth rotation, and it would obviously be visible even with a stationary airplane. The transport wander is the effect of changing the Earth parallel, ie moving closer or far away to/from one of the two poles.
In two words, changing longitude generates the drift, while changing latitude generates the wander. In any case, after a while, the gyro in the airplane is automatically realigned in the IN system.
In my view, the gyro proof is just the smarter version of the Foucault pendulum: and this is the stronger proof of a rotating round Earth, observable by anyone and therefore totally independent from any fakes.