I think there are 2 aspects here. One is that PP has deliberate internal resistance to be able to regulate the mains feeding your equipment. If there wasn’t one, PP would try to regulate the whole network, which is not possible for such a small unit. PSA are constantly trying to improve in this department. So adding a thin core cable pretty much makes things “worse” in impedance department (internal resistance + cable impedance). So better (lower impedance) cable, if can say so, equalizes things to some extent. We are talking ohms and milliohms here.
2nd, I think, cable makes the difference, but not because it is better or worse. It is different. Different construction, metal composition, joints, connections, strands, shape, etc. Different enough to make sonic differences. Whether those sonic differences are better or worse is what makes that cable better for that person, but not necessary better for someone else. Because changing the cable in the wall or inside the PP is rather a specialist job, juggling the user accessible cable is one option to tweak the sound.
For example, I run cheaper thinner cables to my amps, because I find this “sagging” makes the sound smoother and less edgy.