In what sense is the radian a natural measure of angle and the degree an arbitrary measure of that same quantity? Hence what advantages are there in using radians rather than degrees?

The measure of angle in radians is defined as:
angle = arc / radius
ϕ = s / r
Here ϕ is the measured angle, s is the length of the arc and r is the distance between the rotating point and the axis of rotation.
It can be seen from above that the angle is defined using the linear variables “length of the arc” and “the distance between the rotating point and the axis of rotation”, which makes radian as a natural measure of angle.
The relationship to the linear variables is easier to analyze and to get the sense of the motion of the point in terms of linear physical quantities such as linear acceleration, linear velocity etc.
Therefore it is insisted that one must use radians as a unit when the equation involve both linear and angular variables because describing the angular variables in terms of a linear term makes the calculation and interpretation part of the problem easier.
Using degree as a unit involves complex factors appearing in calculations.