They tended to think that this showed that the planets and stars were carried around on the specified configuration of rotating circles, but all it actually showed was that the points of light in the sky followed the paths the theory predicted.

I've never gotten that impression. Proponents of epicycles were working from the assumption that celestial motion must be perfect, and therefore circular, and so were making the math line up with that. Aside from trying to fit an elliptical peg into a circular hole, they seemed to merely believe that the points of light in the sky follow the paths the theory predicts.

But then, it's been a few years since I've read any of the relevant sources.