I didn't read the entire article in detail, it's been a few years since I've done math at those levels, but it makes sense to me and I have to agree on a few of the points. I agree that using the radius is more intuitive than the diameter, you very rarely hear someone referring to the diameter of a circle.
My problem with it though is that "pie are square(d)" rolls off the tongue much more easily than "two tau are square(d)"