Eh… The solar system is a complex system. Man has tried to understand this system by constructing simple models. A fire god riding his chariot across the sky, spheres of rotating crystal with stars imbedded within them, or masses in motion maintaining the same energy state. They are models.
If a model, given a set of data, results in an observed condition then that model is useful or ‘correct’. If two contradictory models, both using the same set of data, result in the same condition then both are useful. The simpler one is preferred because it’s simple and therefore ‘correct.’
People confuse a useful model (the Earth goes around the Sun) with true (1=1). One plus one equals two is not a model. It is.
The question isn’t which model is correct, but which gives the most useful answer in a given situation using a set of data.
Say you’re a backyard skywatcher. To say the Earth goes around the Sun is good enough. If you’re a navigator making his own nautical almanac it would be wise to include the Moon (as the Earth goes around the Earth-Moon-Sun barycenter.) If you were an astronomer you may want to include Venus, Mars, Jupiter, Saturn and Neptune for the same reasons. If you’re a cosmologist… well, you get the picture.
If, as you say, the simpler answer is most likely then the Sun is a god riding a chariot.