With orbiting bodies, the reason an accelerometer doesn't register the least amount of acceleration in any direction is because an orbit is the balance of all such potential directional forces—via motion. Stop the motion of such a body and the force of attraction will be manifest by the object moving straight toward the larger body it was orbiting just prior to that stoppage.
One of the main reasons given for why relativitistic postulations are needed is that no acceleration in any direction is sensed on or near to an orbiting body. This is apparently why Einstein believed geodesic lines, as a fabric in space, cause orbiting; or at least that such hypothetical lines would play an important role. But if the motion itself is what balances those forces so that virtually nothing is felt from the primary body at the location of the secondary body, then our reason for relying on Relativity to help explain that is removed.
Furthermore, the reason our bodies are pulled toward the Earth the amount of 1 g at sea level is because our bodies aren't traveling with sufficient horizontal type velocity to balance out that pull of attraction that occurs between our bodies and the massive interior of the Earth, so that no force would register. For if the person were to travel at just the right velocity over the Earth's surface, if it had no atmosphere to create drag on that said person (and thereby slow them down), then they would be in orbit similar, in principle, to the Earth or Mercury over the Sun.
In orbits, to prove that attraction force—not geodesic lines of spacetime—is ultimately responsible for what occurs, the fact that bodies in the same solar system (during a conjunction) pull one another off course from having more perfect orbits around their star demonstrates that gravity is a directional attraction reciprocating directly between two bodies. Another contradiction to Relativity is tidal force, which also demonstrates how gravity is a rectilinear (as well as constant) attraction between two bodies.
Even though the reciprocal attraction between the Sun and a planet isn't felt on that planet—somewhat similarly to nothing much being felt when driving in a straight line at a constant speed down the highway—the force of attraction between the Sun and that planet is still there even though no force is sensed by any object, or by an accelerometer, when either have been placed on its surface. Once again, that raw force of gravity between the two major bodies is balanced by the motion (per its particular distance) of the smaller body [while that size relationship of a smaller body circling a much larger one appears to be the most typical situation out there], it always takes enough velocity for the orbiting body to maintain its separation—even as the larger a body's mass might happen to be, the more velocity will be required to keep it aloft... which realization the equivalence principle has been attempting to controvert via two spheres of not enough contrasting mass dropped side by side from a tower, also by a hammer and feather dropped side by side on the moon, which is also not enough contrast in mass to demonstrate what's needing to be learned. In other words, if the moon were moved to 30,000 meters over the Earth, instead of its roughly 239,000 miles up, and it would need somewhere on the order of thirty to fifty times its present rate of speed so that it wouldn't come crashing down with its path intersecting the Earth's.
Meanwhile, Earth wobble was indeed a recognized phenomenon many centuries ago, at least by a few. Nevertheless, it now looks as though Einstein was unaware that the Sun wobbles too. Even so, Sun wobble is the cause of Mercury's orbit anomaly, not effects allegedly cause by Relativity. And regarding that accuracy of the Einsteinian equation relating to Mercury's orbit anomaly: keep in mind that he became quite familiar with the way Newtonian equations didn't quite supply enough to account for that, as well as by how much. That means he had a numerical value target that he could aim for, which realization should erode the value of his prediction capability by quite a lot.