You don't think that the -1 is a bit clunky? e^iτ=1. See https://tauday.com/tau-manifesto:
Geometrically, multiplying by e^iθ corresponds to rotating a complex number by an angle θ in the complex plane, which suggests a second interpretation of Euler’s identity:
A rotation by one turn is 1.
Since the number 1 is the multiplicative identity, the geometric meaning of e^iτ=1 is that rotating a point in the complex plane by one turn simply returns it to its original position.
As in the case of radian angle measure, we see how natural the association is between τ and one turn of a circle. Indeed, the identification of τ with “one turn” makes Euler’s identity sound almost like a tautology.