It's simpler than that. Draw a circle of radius 1, and draw two lines through the center. The distance along the circumference between those lines is the angle

It's simpler than that. Draw a circle of radius 1, and draw two lines through the center. The distance along the circumference between those lines is the angle between them in radians. If you really want to remove the multiplicative identity as a magic number, you can define the radian angle as the ratio of the subtended circumference over the radius. IMO this is at least the most accessible argument for why radians are special, and while I don't pretend to understand complex exponentiation, I expect it's the root of why other math involving radians turns out nicely.