I do not quarrel with your assertion that it is possible to derive equations that describe planetary motion, and apparently we agreed that these equations approach accuracy as the number of terms approach infinity. I said that in sloppier language, or meant to. It also appears that modern astrodynamics may use epicycles, to my surprise. I found a research paper:

https://authors.library.caltech.edu/24754/"This paper presents a modern treatment of epicycle theory, which is an exact series representation of Keplerian motion, and uses that theory to develop the first analytic method for analyzing the higher order dynamics of the LISA orbits. LISA, the Laser Interferometer Space Antenna mission, uses a constellation of three spacecraft in heliocentric space and takes advantage of particular solutions of the Clohessy-Wiltshire equations."

So I was wrong, epicycles are used by modern astronomers, but I don't think much. Note that the author is confirming the equivalence to Kepler.

From the wiki page on deferent and epicycle:

https://en.wikipedia.org/wiki/Deferent_and_epicycle"Epicycles worked very well and were highly accurate, because, as Fourier analysis later showed, any smooth curve can be approximated to arbitrary accuracy with a sufficient number of epicycles. However, they fell out of favor with the discovery that planetary motions were largely elliptical from a heliocentric frame of reference, which led to the discovery that gravity obeying a simple inverse square law could better explain all planetary motions."

This is the overall approximate truth I was under the impression of, and I think it is true that most astronomers today do not use epicycles. In math, there are often different approaches that yield the same results, and apparently, epicycles helped at least one modern astronomer, who, as I mentioned, said it was equivalent.

And it remains true that epicycle equations are produced by taking data and analyzing it without relation to underlying physical theory. It is a case of, here is an equation that produces the curve described by the data, while Newton/Kepler says here is mass and position and velocity and equations describing experimentally confirmed forces, and those equations accurately predict observed apparent planetary motions produced by heliocentric RE astronomy.

Case in point, astronomers observed slight variations in the predicted orbits of known planets, and used the Newtonian equations within RE heliocentric solar system to predict the existence of Neptune. They looked where they calculated it should be, and there it was. Epicycles could not do that.

But ... that was not the question of the original post. The original question is whether Newton/Kepler equations "worked", are they consistent and predictive. Per the above articles and general knowledge, yes they are. My point is that the motion of planets on FE dome may be described and even predicted by epicycles, but they don't explain why planets appear to make little loops, slow down, go backwards (planets to ancient Greeks: wandering stars). Kepler/Newton/RE are also consistent and predictive, as well as explanatory. If FE is true and planets are moving around on the dome, it is amazing to me that there is a 3d explanation that matches with known laws of physics. What a coincidence! Heliocentric Kepler/Newton RE solar system has the same appearance as the FE dome.

If you take FE as true, you don't know the how or why of the appearance of planets and other heavenly bodies, nor day and night, different stars in the southern and northern hemispheres, etc. You don't know why when it is sunset in Denver, people looking at the dome directly over Denver from St Louis see dark sky with stars, while at the same time people in Salt Lake City look at the same spot on the dome and see light blue with no stars. RE explains this consistent with known physical laws. FE requires some complex, speculative, unproven explanations, or has no explanation at all. "Some people have proposed a model ..." is not proof, not complete, not consistent.

Please point me at the place in the wiki that explains how people can see such different things when looking at the same spot on the same dome, and I will rebut it. I have looked (and asked), and I haven't found it.