How gravity works and how whatever-swings-the-FE-sun about works are both unknowns, sure. However:
The hypothesis of universal gravitation immediately and totally predicts all celestial and orbital behaviours within a very wide range of speeds. It does so precisely enough to enable a vast new tract of science and technology. Outside those ranges, special and general relativity can be used to make precise predictions, these are precise enough to predict celestial motion right out to almost the largest of scales. With relativity comes not only a working model of most of the heavens but global timekeeping to a trillionth of a second and all the science and technology that enables.
The implied question, which is a fair one, is: what predictions can be made with the hypothesis that the sun moves "mysteriously", or that its orbit "changes".
Or in other words: is there an FE model which can predict the observed movement of a celestial body across the sky? Or do the movements (which are all predicted by RE models) just get filed under "mysterious"? And finally - is that really a satisfactory answer?