If it says the same thing as SR, why do you prefer it to SR?
It doesn't say the "same thing." I'll come back to this.
That condradicts something you posted above from wiki, about the two theories having the same mathematics and empiric consequences.
That's only true, if at all, in any one case comparison. However, the Hafele-Keating experiment involved 3 clocks as follows:
1. A clock at a naval station in Maryland,
2. A clock on a plane which took off from there and flew west, and
3. A clock on a plane which took off from there and flew east.
The planes flew at the same speeds, altititudes, etc. SR says the speed of light is constant, regardless of the motion of either the emitter or receiver of the light, that all time dilation is "reciprocal," etc.
If, using SR, and using the naval station as your chosen frame of reference, you calculate the time differences you would predict that each clock on the planes would slow down by an equal amount, relative to the "stationary" clock at the naval station. If you used either other plane as your chosen frame of reference, you would get different (conflicting) answers.
The absolute differences in accumulated elapsed time were measured, after the flights were completed. As it turned out, the plane flying west showed the least elapsed time, the "stationary" clock at the naval station was "in the middle" and showed more time elapsed than the westbound plane, and the eastbound plane showed the least time elapsed. Translated this would mean the eastbound clock moved the slowest and the westbound the fastest, amongst the three.
There was one, and only one, frame of reference which could be used which would accurately predict the amount of time that elapsed on each clock, and that was a non-rotating earth-centered inertial frame of reference. Using such a preferred frame requires you to posit absolute simultaneity, with all differences in time being slowed more or less WHEN compared to the "master clock" at the ECI.
Enough for one post.