Which shouldn't come as too much of a surprise because the radius is the smallest amount of information that determines what a circle is, as well as the basis for how we define radians.
The radius is not the smallest amount of information which determines a circle. The radius and origin can determine a circle. So can the diameter and origin, or the circumference and origin, or the area and origin. Don't forget your geometry.
Sure, you can write any in terms of any of the others, but the radius is the smallest: both in terms of absolute value as well as dimensionality.
As for dimensionality, how are you using the word? No matter what, we need 3 numbers to define a circle in R^2. Two numbers define its position and one number defines it size. I don't see what you're getting at. If you mean that because radius is a measure of length (one dimension) and area is a measure of area (two dimensions) then I have two questions: 1) Why does that matter? Its still just a single number. 2) Even if it does matter, why is radius more fundamental than diameter.
Was two millennia not long enough? We could take another couple centuries, I guess. I don't think we're ever going to get a new answer for the ratio of the circumference of a circle to the diameter of that circle, though.
> The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful.
Frankly, I don't know what this means. I've read it several dozen times, and each time, it seems increasingly more inane. I can't help but wonder what you would say about the derivative (and antiderivative) of e^x; would you complain that it's not meaningful? Does it need more coefficients? More exponents?