So swapping your square with circle and centre, "4" with "3.14"??
Staring at the patterns too much, we lose the glimpse of the "whole." We must see the trees and jungle both at once. It is all about training the vision.
We must revers decades of training our eyes to ignore the essentials.
You: "There’s a lot more we can do with PI than with 4… I think"
I understand , you can role and wheel, but in square all the pointes, angles of attachment are fulfilled. You can not divide a surface or a volume to circles, only to squares. PI always IS going to be missing that 4-3.14= '0.86' portion. (about 21.5 %)
Also square is more "Feng Shui" (not for no reason)