If you want to use pi, then you should be using the angle subtended by the length of the diameter, not the radius, as the basis of your angle measurements. Be consistent.
The reason why radians are what they are is because then the length of the arc (of a unit circle) that subtends the angle R is exactly R. That's really convenient. (You could use diameter, but in real work you've got the radius far more often than the diameter.)
And any other ratio (especially tau!) doesn't satisfy the Eulerian identity, so can't be nearly so beautiful.