Eh, you can find plenty of cases where tau is just as awkward as pi is elsewhere. Right off the bat, the area of a circle becomes more awkward with tau, becoming (tau*r^2)/2, and in general, the volume of an n-ball gains weird powers and roots of two in its denominator as n increases if you switch to tau. In general, I don't think you can really claim either one is "more fundamental". It's just a matter of framing.
Actually, no: that Tau-centric area formula you gave derives naturally from taking the integral. Your example actually fits the expectation you have from what you learned in Calculus I. You should _expect_ that 1/2 scaling to be there.
If it seems awkward to you, it's only because of a lifetime of seeing it done in terms of pi.
You can argue that having an extra number to juggle around is somehow less awkward because, under specific and subjective criteria, it’s “expected”, but given that the whole hook for tau is “we keep having to put a multiplier of two everywhere”, I don’t find it very compelling.
I can also make arguments that pi/2 would have been a better constant from a teaching perspective. The pi/2 version of the Euler identity, for example, would give you all the tools you need to link complex multiplication to rotation.
But at the end of the day, the choice of multiple used for the constant is a convention. None is going to be ideal in every case, and no math fundamentally changes because of a particular choice. Trying to argue for a change in convention at this point is just silly.
The entire tau manifesto is basically an exercise in how you can come up with rationalizations for just about any aesthetic preference, and the “area of a circle section” is a perfect example of how far you can go with the gymnastics.
It's not really about being awkward (that's a tell not the motivation), it's about basing on a radius or diameter: which is more fundamental? Or the arc length of a unit circle or half a circle, which isn't an arbitray formula it's the definition.
