[Hondor]

You'd lose the definition of angle as arc length / radius, which would make your proposed unit just as arbitrary as degrees - most formulas involving angles would need to contain a conversion factor.

You'd also lose the small-angle approximations sin(theta)=tan(theta)=theta which in my field are used extensively to convert nonlinear to linear equations.

Nonetheless, your proposed unit already exists and is called tau. Or write it as 2pi if you want to be more easily understood.

[majewsky]

The most relevant point here is that the natural choice of units is very subjective and depends on the task at hand. For example, particle physics uses "natural units" where all units are powers of gigaelectronvolts: https://en.wikipedia.org/wiki/Natural_units#.22Natural_units...