The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series.
There is only one consequence of those series that matters in practice, which is that when the angles are expressed in radians, for very small angles the angle, its sinus and its tangent are approximately equal.
While this relationship between small angles, sinuses and tangents looks like an argument pro radians, in practice it isn't. There are no precise methods for measuring an angle in radians. All angle measurements are done using an unit that is an integer divisor of a right angle, and then the angles in radian are computed using a multiplication with a number proportional with the reciprocal of Pi.
So the rule about the approximate equality of angles, sinuses and tangents is at best a mnemonic rule, because to apply the rule one must convert the measured angles into radians, so no arithmetic operations can be saved.
"Turns" generalize perfectly to higher dimensions.
To the 3 important units for the plane angle, i.e. right angle, cycle and radian, there are 3 corresponding units for the solid angle, i.e. the right trihedron (i.e. an octant of a sphere), the sphere and the steradian.
The ratio between the right trihedron and the steradian is the same as between the right angle and the radian, i.e. (Pi / 2).
The ratio between the sphere and the right trihedron is 2^3, while that between cycle and right angle is 2^2. In N dimensions the ratio between the corresponding angle units becomes 2^N.
Moreover, while in 2 dimensions there are a few cases when the radian is useful, in 3 dimensions the steradian is really useless. Its use in photometry causes a lot of multiplications or divisions by Pi that have no useful effect.
There is only one significant advantage of the radian, which is the same as for using the Neper as a logarithmic unit, the derivative of the exponential with the logarithms measured in Nepers is the same function as the primitive, and that has as a consequence similarly simple relationships between the trigonometric functions with arguments measured in radians and their derivatives.
Everywhere else where the radian is convenient is a consequence of the invariance of the exponential function under derivation, when the Neper and radian units are used.
This invariance is very convenient in the symbolic manipulation of differential equations, but it does not translate into simpler computations when numeric methods are used.
So the use of the radian can simplify a lot many pen and paper symbolic transformations, but it is rarely, if ever, beneficial in numeric algorithms.