Radians vs. degrees isn't notation, it's a convention. You even say the reason why it is the convention. That multiplying the radius by the radians gives you the circumference. It is the only representation of angles with this special property. I mean, why should 360 represent 1 rotation? Why not use rotations itself? That way 1/4 is 1/4, 1/8 is 1/8, and so forth.