When we calculate center of mass using rectangular coordinates, we find the average values in each coordinate. Obviously we can't do this very same thing in polar coordinates: if we integrated a constant density on a disk, we'd end up with a positive "average $r$" value, which doesn't make any sense.
What is the fundamental reason why the averaging coordinates works in the former case but not the latter? I suppose it showcases a fundamental difference between rectangular and spherical coordinates.