The Haefele-Keating experiment, conducted in 1971, was designed to test the predictions of time dilation in special relativity. While it is true that the clocks used in the experiment were not in inertial frames of reference, the experiment itself still provided empirical evidence consistent with the predictions of special relativity.
In the Haefele-Keating experiment, atomic clocks were flown around the world in commercial airplanes traveling in different directions and at different speeds. By comparing the readings of these clocks with reference clocks on the ground, the researchers aimed to observe the effects of time dilation predicted by special relativity.
Although the clocks were not in inertial frames due to their motion in the airplanes, the experiment is still considered a test of special relativity because it examined the influence of relative motion on time. Special relativity predicts that time will dilate for objects in motion relative to an observer in an inertial frame of reference. In this case, the observers on the ground were considered to be in a nearly inertial reference frame, as the motion of the Earth's rotation was negligible compared to the speed of the airplanes.
The Haefele-Keating experiment demonstrated that the atomic clocks on the airplanes, due to their relative motion, experienced time dilation compared to the reference clocks on the ground. The observed time differences were consistent with the predictions of special relativity.
It's worth noting that while the center of the Earth (a non-rotating frame) can be considered a privileged reference frame for certain purposes, it is not necessary to invoke it specifically for the Haefele-Keating experiment. The experiment's design allowed for comparisons between the moving clocks and the clocks on the ground, providing a test of time dilation effects in special relativity.