I remember introducing someone to the concept of variance in a set of data and I used a very similar approach. Variance seems like an arbitrary (but obvious) definition but in fact it can be derived from first principles by just looking for the simplest possible function that firstly has some dependency on the difference between values and the arithmetic mean, secondly has the property that it is independent of whether the differences are positive or negative (to the right or left of the mean) and thirdly that it does not depend on the size of the data set (i.e. duplicating each member of a data set would leave the variance unaffected). When you consider each of these, the equation for variance arises very naturally.

Arithmetic difference satisfies for first property

Squaring each difference satisfies the second property

Taking the arithmetic mean satisfies the third property

Var(X) = E[(X - mean)^2]