Although Einstein didn't live long enough to see the key solution to his own theory of gravity, dipole gravity is certainly a result of special relativity and the equivalence principle which are the fundamental building blocks of general relativity. The equivalence principle predicts that when there is an acceleration of mass, it must generate gravitational effect associated with it. And the object in rotational motion in mechanics is basically in a state of continuous acceleration in the sense that the motion requires the continuous change of the directions of the mass element within the rotating body.
But it was not clear in regard to how that works, in what way and through what kind of mechanism to cause a realistic gravitational field.
The analogy with electro-magnetism certainly failed. Rotating donut shaped object didn't create gravito magnet. Lens-Thirring's force from the rotating spherical shell indicated there is a remnant of the acceleration of the mass induced force near the center of the sphere but the interpretation and the sign of the force were incorrect. The radial component of the force was interpreted as the centrifugal force in accordance with Mach's principle and there was no explanation for the harmonic force along the axis of the rotation. And none of these forces seemed to give any clues to the existing cosmological problems. And it didn't encourage any further investigation into the subject largely because of the horrendous difficulties in analytically calculating all the details of the gravity effect from the circular motion of the spherical object. And the dipole term from the linearized theory was simply interpreted as meaningless and it was declared that there is no such thing as dipole gravitational moment as a physically meaningful entity.
Thus, the question remained a fundamental mystery of general relativity. How the circular motion of the mass manifests itself into a verifiable gravito magnetism?
Even if the rotating frequency and the tangential velocity of the object may be considered crucial to the effect, it was not obvious how it should be incorporated into the final form of the gravito magnetism. The most crucial and final stumbling block may have been Newtonian mechanics itself since there is no such thing as the shift of the center of mass by a fixed definite quantity depending on the speed of the rotational motion in Newtonian mechanics. It simply won't happen. Of course, this can be used to perfectly justify the claim of the non existence of the gravitational dipole moment if and only if one assumes that Newtonian mechanics is the flawless and complete theory of the mechanical universe.
Dipole gravity(gravito magnetism) is a plain and inevitable solution to general relativity that has been sought but eluded the researchers in the field for a long time. Clearly it was not obvious to Einstein himself. But there is no doubt that it is still one of his most triumphant legacies.
He would have been very happy to see the solution of his own theory explaining the mysteries of the universe ranging from the jet phenomena to the dark matter problems which did not emerge as the prominent cosmological problems at his time.
We can learn here the pattern regarding on how the old and new principles of physics exchange and compete to evolve into a new principle to teach us something new. When there is a new principle which is larger and incorporating the old principle, there is a crucial and unique mechanism that shows the clear conflict between the old and new principle at which point one is forced to choose one and thereby face the demands of the modification of the old principle to make it compatible to the new one.
In any circumstances, the final result is that it forces us to abandon certain features of the old principle. This happens in a continuous fashion in the history of mankind and that is how the secret of the nature is continuously made comprehensible.