At some point I'd like to carefully study the history of that early era of science because I don't know it as well as I'd like. But I believe I understand that the epicyclical model (and it was a model, rather than a theory) did not in any way depend on geocentrism. For one thing, Coppernicus' model itself, while heliocentric, retained the epicycles of the earlier, geocentric model. Instead, the assumption on which the epicyclical model depended was the shape of the planets' orbits and of the planets themselves, which were considered to be necessarily circular, and spherical, respectively. I think this had to do with assumptions about the geometric perfection of the universe, as a creation of the gods. In any case, assuming that planetary orbits were circular an explanation was needed for the apparent "retrograde" motion of the planets (meaning it looks like they double back and turn against their original heading). Explaining this apparent motion was why epicycles were hypothesised in the first place.

The first time this necessarily circular model was abandoned was with Kepler's laws of planetary motion, which correctly identified the motion of the planets as elliptical, that for the first time explained their apparent retrograde motion without the need for epicycles. Then Newton's theory of universal gravitation explained how the planets could possibly be moving on elliptical orbits. In fact, I believe Newton's theory of universal gravitation explained how the planets could be turning around the sun without crashing down, despite not having anything to hold them up. I think this was the first big mystery that the ancients tried to answer- hence the name of "firmament" for the universe.

And Newton's theory was not the end of the story of course.