> Relativity doesn't say that both an Earth-centric and solar-barycentric models are equally correct

I'm fairly sure it does. Of course, you'll get the same answer either way - the earth moves in an elliptical orbit around the sun and contributes a tiny amount of the total inertia of the system. But you can use either the sun or the earth as your frame of reference to calculate this.

It's explained quite well on wikipedia so I'll just quote that here:

> Albert Einstein and Leopold Infeld wrote in The Evolution of Physics (1938): "Can we formulate physical laws so that they are valid for all CS (=coordinate systems), not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any CS. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS could be used with equal justification. The two sentences, 'the sun is at rest and the Earth moves', or 'the sun moves and the Earth is at rest', would simply mean two different conventions concerning two different CS. Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!"[47]

> Despite giving more respectability to the geocentric view than Newtonian physics does,[48] relativity is not geocentric. Rather, relativity states that the Sun, the Earth, the Moon, Jupiter, or any other point for that matter could be chosen as a center of the Solar System with equal validity.

> Relativity agrees with Newtonian predictions that regardless of whether the Sun or the Earth are chosen arbitrarily as the center of the coordinate system describing the Solar System, the paths of the planets form (roughly) ellipses with respect to the Sun, not the Earth. With respect to the average reference frame of the fixed stars, the planets do indeed move around the Sun, which due to its much larger mass, moves far less than its own diameter and the gravity of which is dominant in determining the orbits of the planets (in other words, the center of mass of the Solar System is near the center of the Sun). The Earth and Moon are much closer to being a binary planet; the center of mass around which they both rotate is still inside the Earth, but is about 4,624 km (2,873 mi) or 72.6% of the Earth's radius away from the centre of the Earth (thus closer to the surface than the center).[citation needed]

> What the principle of relativity points out is that correct mathematical calculations can be made regardless of the reference frame chosen, and these will all agree with each other as to the predictions of actual motions of bodies with respect to each other. It is not necessary to choose the object in the Solar System with the largest gravitational field as the center of the coordinate system in order to predict the motions of planetary bodies, though doing so may make calculations easier to perform or interpret. A geocentric coordinate system can be more convenient when dealing only with bodies mostly influenced by the gravity of the Earth (such as artificial satellites and the Moon), or when calculating what the sky will look like when viewed from Earth (as opposed to an imaginary observer looking down on the entire Solar System, where a different coordinate system might be more convenient

https://en.m.wikipedia.org/wiki/Geocentric_model#relativity

See in particular this sentence:

> It is not necessary to choose the object in the Solar System with the largest gravitational field as the center of the coordinate system in order to predict the motions of planetary bodies, though doing so may make calculations easier to perform or interpret.

That's exactly the point I made above.