I don't understand what you're trying to get at here. Surely the shortest euclidian distance between X and S is obtained by setting s=0, which has nothing to do with the mean.

In nabla9's example, S would be the "vector discrepancy". If you want to minimize the discrepancy of s and X (NB. s and X, not S and X, I think this was a mistake), you want this vector to be as small as possible. In other words, you want the magnitude of the S vector to be minimal. This magnitude is given by the norm of the vector, sqrt(sum((S_i)^2)), which will reach a minimum when sum((S_i)^2) reaches a minimum.

Ok, but how does that provide intuition for sum((S_i)^2) reaching a minimum at the mean of the x values?

Can take the derivative and set it to zero and will get the definition for mean.

Sure, but that's not intuition.