I actually disagree that that is nicer. e^(ipi) = -1, which means you can square both sides to come to the also factual statement, e^(itau) = 1. However, if you only knew the latter you'd be wondering whether e^(i*pi) would be 1 or -1.
e^(tau*i/2)+1=0
e^(i*tau) = 1
Considering that, we'd need to state Euler's identity as: e^(i * pi/2) = i, but again we'd know nothing about fractions of that exponent. And there we go!
The point is, Euler's identity is a nice property of the definition of complex numbers, but in itself, not too generic. That's why OP's form, e^(i*tau) = 1, would do just fine.
Edit: fixed asterisks.