Earlier quoted context omitted.
I think most simply, the entire solar system revolves around the sun and not the earth.
Isn't that simply begging the question though? If you adopt a geocentric reference frame, then the Sun appears to rotate around the Earth, and all the other planets also appear to follow epicyclic paths around the Earth. You can predict the motions of every planet and star equally well, the equations are just different. So again, why is one more 'correct' than the other, besides calculational simplicity?
The larger issue you raise is one that was especially prominent in mid-20th century philosophy of science, specifically in Thomas Kuhn's concept of a "paradigm shift". A radically new scientific theory does not displace an existing theory because it is more "correct", since ad hoc modifications (e.g., epicycles) can always be made to the existing theory to account for new evidence that appears not to fit. However, at some point the existing theory will come to have so much baggage, and the new theory will explain the world so much more elegantly (e.g., with calculational simplicity) that the new theory will overthrow the old one and become the new "paradigm". Note that this is not because the old theory could not accommodate all the facts; any old theory can if you modify it enough, but at some point when you have to make all sorts of strange adjustments to the existing (old) theory it just becomes wildly implausible when compared with the new theory. At some point scientific consensus changes and the the new "paradigm" is accepted. You can read more about this at Wikipedia:
https://en.wikipedia.org/wiki/Copernican_Revolution
https://en.wikipedia.org/wiki/The_Structure_of_Scientific_Re...