Oh. I think we're beginning to drift a bit here. (Surprise surprise for PF)
Choosing the CM of multiple bodies would be fine in some circs for calculations and it could have the advantage that the Potential wouldn't actually be at -∞ (there would be a number of potential wells around it, corresponding to the various bodies involved).
Going back to the original question and the original situation - you can choose the CM of a planet / star as a reference but the 1/r law stops applying once you get below the surface and the actual depth will depend upon the density distribution within the planet. This would hardly be practicable. You could use the surface - at the mean / max / min radius point. This wouldn't be easy to define either. Or you could use ∞. To my mind there is no contest here and I am not at all surprised that it is ∞ that's used pretty much universally (I probably mean that literally, too).