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...
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.