I don't think that the analogy holds as I'll explain below.
I've seen it argued that it WOULD make a massive impact on the way one learns trig. It is more intuative and elegant. We use the circle constant by the radius rather than the diameter, so why not use the circle constant that is based on the radius rather than the diameter?
Tau is conceptually more intuative in that it represents one whole rotation(or *gasp* turn). It makes learning trig easier as it doesn't really make a whole lot of sense to learn angles in terms of half rotations.
Oh, and it's awesomely elegant for one reason:
[math]e^{{\tau}i}=1[/math] ties together e, the circle constant, i, and 1 without manipulation. Well, that actually has a little bit of manipulation as it really is [math]e^{{\tau}i}=1+0[/math] which ties together e, the circle constant, i, 1, and 0 far more naturally than if you would use pi.
So, like I said, your analogy doesn't quite hold since changing the letter e is neither conceptually more simple nor more elegant. Having said all that, however, it is still a simple substitution at the end of the day.