The Hafele-Keating experiment is often analyzed and discussed using the reference frame of the rotating Earth, which can be considered an inertial frame for short periods of time. This reference frame is commonly used because it simplifies the calculations and is a convenient approximation in this context.
However, it is indeed possible to analyze the experiment from the reference frame of the airplane, which is a non-inertial frame due to its motion and acceleration. Performing the analysis from a non-inertial frame introduces additional complexities and requires considering the effects of acceleration and non-inertial forces.
In practice, analyzing the Hafele-Keating experiment from a non-inertial frame would involve considering factors such as the velocity and acceleration of the airplane, the changing gravitational potential due to Earth's rotation, and the associated gravitational time dilation effects.
While I couldn't find a specific analysis of the Hafele-Keating experiment from the airplane's non-inertial frame, it is important to note that special relativity is a well-established theory that provides consistent predictions and explanations for a wide range of phenomena. The Hafele-Keating experiment, in conjunction with numerous other experiments and observations, supports the validity of special relativity.
The choice of reference frame in analyzing the experiment is a matter of practicality and convenience. The rotating Earth frame is often used because it simplifies the calculations while still providing accurate predictions consistent with the principles of special relativity.