How does that answer the question?It's true that the center of gravity is a mean, but the moment of inertia is not a variance. It's one thing to say something is "proportional to a variance" to mean that the constant is 2 or pi, but when the constant is the number of points, I think it's missing the statistical point.
But the bigger problem is that these are not statistical examples! Means and sums of squares occur many places, but why are they are a good choice for the central tendency and the tendency to be central? Are you suggesting that we think of a random variable as a physical rod? Why? Does trying to spin it have any probabilistic or statistical meaning?
Mean and variance are closely related to center of mass and moment of inertia. This is good intuition to have, and it's statistical. The only difference is that the first two are moments of a probability distribution, and the second two are moments of a mass distribution.