By "worst possible", in the context of this puzzle, I meant the distribution of points that maximizes the distance between any two points, which is the same as your definiton. Maybe I didn't describe the distribution very well, but that's why I attached the image.
I'm not sure I follow you here completely, but some things that work in 2 dimensions don't work once you apply them to a 3-dimensional problem.
I agree that I haven't provided a formal proof of my solution, but that's all I got for now.