> you're not incorrect if you pick a frame that puts the moon in the center or any other point

I think you're conflating different things. Yes, we can predict motion from the view of any point and that would likely involve epicycles. But that's not what people were after. Causality matters, a lot. It's kinda the whole point of physics (and what makes it so expressive and powerful).

What classical relativity does not allow you to do is place yourself at an arbitrary point and call that the center of mass of the system (i.e. things orbit around you). That's not invariant. It's a very different thing than predicting the motion of objects from your vantage point.