Radians are generally accepted as natural angular units of measure because they are dimensionless and lead to simple derivatives and series expansions. However, in the modular arithmetic of wavelength, frequency, and other full circular multiples, such as in Fourier analysis, multiples of $2\pi$ abound.
The use of 'turn' (the angle of one full circular rotation) has an advantage that its natural full-circle modulus is simply the floor operator.
Another potential angular measure would be the quarter turn (qturn). In it, the quadrants of a unit circle are represented as integers, with a full circle being 4 qturn. In it, many significant angles are rational numbers. And trigonometric reductions to the first qturn become simple operations involving integers.
Are there any other advantages of using turns or qturns to measure angles? Do they have significant disadvantages?
The parametrization of the unit circle by radians is $t \mapsto \exp(i t)$ and so has speed $1$.
The parametrization of the unit circle by turns is $t \mapsto \exp(2\pi i t)$ and so has speed $2\pi$.
As you can see, there is no escaping $2\pi$.