The radius is only more fundamental to the notion of a circle because of the modern definition of a circle (set of all points exactly r away from the given point). One could also use a definition like: "Continuous set of points such that each is distance d from exactly one other, and further from none." The modern definition is convenient because it's a special case of a convenient definition of ellipse, but there's nothing terribly fundamental about it.

And yes, every time you've done math with radians, that problem has haunted you. That's because we decided to use radians instead of "diameter-ans", to keep with the above definition.