In LIGO, we detect the far away distortions propagating through our rulers and the differential effect of those waves.
But Michelson-Morley failed because the ruler distorted synchronously with the signal we tried to measure — since we mistakenly believed EM and matter were different fields.
General-relativistic effects are relevant for the outcome of the Michelson-Morley experiment in the sense that the general theory of relativity is a generalization of the special theory of relativity. :-)
Otherwise: the additional predictions that the general theory of relativity makes about gravity, curvature of space etc. are not relevant for the outcome of the Michelson-Morley experiment.
You seem to think I’ve said something very technical, while I’ve merely pointed out a well-known effect of Lorentz invariance and interferometry: you can’t detect an “aether” signal coincident with the detector, because the detector is distorted synchronously.
Michelson-Morley was never going to work because it relied on the assumption matter wasn’t subject to aether wind — which turned out to be wrong, as matter is subject to relativistic effects and fields unify. You needed that independence to generate the differential the interferometry amplified.
By contrast, LIGO measures an effect far removed and not co-moving with the detector. So it works.