Not exactly, Shewhart's arguments fundamentally are about the utility of the mean and standard deviation as useful estimates regardless of the underlying distribution. In fact he states in his book:
"...we can make use of the average and standard deviation of a statistic, even when the underlying distribution is not known."
He uses Tchebycheff's theorem to support establishing limits that reasonably identify "when trouble exists" and empirical evidence regarding economy of concluding there is trouble when there is not.
See previously referenced: Shewhart, Walter A. (1931) “Economic Control of Quality of Manufactured Product”, D. Van Nostrand Co., NY, Chapter XIX, p.275-278
"All models are wrong, some are useful" G.E.P. Box