Think of yourself as being attached to the end of a rope that is nailed to the ground. Obviously as you move, an angle would be swept out. One way to measure the angle would be to measure the proportion of the entire circle that has been swept out. That is the degree measurement, or the pivotal angle.
But another way is to measure the distance you have walked around the outside of the circle, so the arclength. Obviously for different lengths of rope, you would walk a different distance on the arc even if you do sweep out the same angle. So this arclength measurement depends on the length of the radius. So we use our radius as our unit of measurement, and ask ourselves "how many lengths of the radius have we walked along the circumference?" This is what a radian is, a length of the RADIUS on the circumference.
Since every circle has a circumference of , that means that there are exactly lengths of the radius on the circumference, or if you like, radians in a circle.
So the reason we don't use is because that gives us the length of the circumference. But for an angle, all that we want to know is how many lengths of the radius there ARE on the circumference.
Does that make sense?