> Why do we insist that we are moving around the sun, and not the other way around if everything is relative?

Because it makes the math easier - that's all. If you fix your frame of reference to the Sun, the orbits of all other bodies become almost perfectly elliptical. If you try to pin the reference frame to any other body, you end up with complicated curves. But they model the same thing[0]. So the whole thing about "Earth orbits the Sun" is that it gives same results, but is much easier to work with.

Technically, for the easy math, the point you're after is the barycenter[1] of the Solar System - the center of mass, which, per Newton's First Law, can be used to center the reference frame, because it's not accelerating[3]. As it turns out, the barycenter of the Solar System spends most of the time within the volume of the Sun[2], and otherwise is very close to it. So for most calculations, you may just as well pin the reference frame to the Sun.

And then, when you fix your sights at the barycenter, you'll notice the movement of celestial bodies fall out pretty much straight from joining Newton's Second Law with the Law of Universal Gravitation - m₁a₁ = Gm₁m₂/r². Your model simplifies - you now realize the movement of celestial bodies is governed by the same laws movement on Earth is, and all the complexity of geocentric model was caused by needless coordinate transformation, due to a bad choice of the reference frame.

Also worth noting that historically, humans have developed the geocentric model to a very impressive level of precision - to the point that the "upstart" heliocentric model initially was worse at predicting movement of planets. It took some extra insights for the heliocentric model to beat the old ones[5] - and only then Newton came along, and people connected the effect with the cause.

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[0] - If I recall correctly, if you were to take the path of a planet in a geocentric model and do a Fourier transform on it - that mathematical operation which represents a function of time as a possibly infinite sum of sine waves - you'd notice that the path of your planet is essentially a sum of two periodic functions. One would correspond to the movement of Earth around the Sun, the other to the movement of the planet around the Sun. This would give you a strong hint that your model is needlessly complicated, and can be recreated using much simpler curves.

[1] - https://en.wikipedia.org/wiki/Barycenter

[2] - https://en.wikipedia.org/wiki/Barycenter#/media/File:Solar_s...

[3] - Only forces from outside the considered system could cause it to move, which we're by definition not considering when talking about our system in isolation. And besides, they add up to negligible amounts anyway. Nice thing about forces in our reality scaling like 1/r^2 or worse[4] is that they very quickly add up to nothing with distance, which makes it easy for us to treat systems as isolated in calculations, and have the results match up to reality with extreme accuracy.

[4] - https://en.wikipedia.org/wiki/Inverse-square_law applies to gravity and electromagnetism; weak and strong forces drop much faster with distance.

[5] - Like using ellipses instead of circles as the fundamental curve, because your competition that used circles moving on circles could just keep adding circles - they were doing a Fourier transform without knowing it, and each circle added a frequency component, increasing the accuracy of approximating the actual ellipse.