> Mathematically, in a two-body system, there's no actual difference between saying body A orbits body B or saying body B orbits body A, so in some sense, it's not even wrong.

This isn't what the geocentric model claimed, though. It went beyond just a choice of reference frame, which as you say, you can do in math, or physics.

For a start, the geocentric model claimed a physically preferred reference frame, which already directly contradicts the coordinate relativism you described. In that sense, it was wrong.

Beyond that, it proposed a mathematical model based on epicycles, a model which was eventually falsified due to many failures to match observation. In that sense, it was also wrong.

These points also contradict your other claim:

> Modern astronomical ephemerides use the Tychonic model--the sun is modeled as revolving around the Earth, the other planets as revolving around the sun.

This is misleading at best. The ephemerides you mention are based on modern Newtonian many-body physics, but they do a coordinate transform on the results to express them in a way that's convenient for Earth-bound observers.

This is not "using the Tychonic model" in any meaningful sense. It's using a correct coordinate transform that is equivalent to the overall coordinate system that Tycho tried to use, but failed to get right. It doesn't rely on any aspects of Tycho's model, because that model was largely invalid, and would not produce correct results.