How to get rid of shading problems with an all-quads-sphere from a cube?

The retopology of the grease fitting sphere reminds of the shape I get by applying the “To Sphere” tool on a subdivided cube.  If I use this approach I can clearly see (with metallic matcap and smooth shading) distortions at the positions where the original cube had its corners. Is there a way to get rid of this? Maybe an all-quads-sphere made from a UV-sphere with the pole triangles merged to quads would be better?

Here is my file where I also tried out the “Shrinkwrap” method for cuts on curved surfaces.