I'm not sure I follow. Multiplying through by two to get tau would simply move the nested square roots from the denominator to the numerator in the relation with the golden ratio (as is explained in the link).

I certainly don't see any step in the outlined derivation that gains or loses the ability to be intuited simply due to a factor of two.

But... why? If you wanted to express their equation in terms of tau (that is, 2pi), you could just set the first term of the right hand side to 10/phi instead of 5/phi. In fact, throughout their derivation there are points where a tau-based version would be a bit cleaner (though of course there are other points where tau might be a bit messier).

i can't tell you how many failed trig tests happen as a direct result of a non-intuitive unit circle defined in terms of Pi [1].

Pi is fundamentally wrong. A circle is defined by a point and a radius. Diameter is just something that has always been easier to measure, so now we're stuck with it :( The fact that Tau makes some formulas less "elegant" is irrelevant.

[1] http://www.shelovesmath.com/wp-content/uploads/2012/11/Unit-...