The Haefele-Keating experiment was a test of time dilation predicted by special relativity, and it involved comparing the behavior of atomic clocks at different altitudes in Earth's gravitational field. While it is true that special relativity is formulated for inertial reference frames, the experiment can still be understood within the framework of general relativity, which encompasses both special relativity and gravity.
General relativity is a theory that describes the effects of gravity on the curvature of spacetime. In the case of the Haefele-Keating experiment, the clocks on the aircraft were subject to different gravitational fields compared to clocks on the ground. General relativity allows us to consider the effects of gravitational time dilation on the behavior of clocks in different locations.
The special privileged reference frame in this experiment is the frame of an observer at rest on the Earth's surface, far away from any gravitational sources, where the gravitational time dilation is minimal. By comparing the behavior of clocks in this reference frame to the clocks on the aircraft at different altitudes, the researchers were able to observe the predicted time dilation effects.
It's worth noting that the Haefele-Keating experiment was not conducted to test the validity of special relativity directly but to test the specific predictions of time dilation arising from general relativity. While special relativity is indeed formulated for inertial reference frames, the experiment was designed to investigate the gravitational time dilation predicted by general relativity in the presence of Earth's gravitational field.