This is not why it works. The James-Stein estimator applies in the case of independently distributed Normal variables with equal variance, so the individually optimal estimators for each parameter have the same MSE.
I think it is impossible James-Stein estimator of the whole sample can outperform all of the three MLE for each particular. Not that I would know of a way to generate a particular from a joint estimator.
I think I may have misinterpreted your original comment. It's true, as you say, that in the Stein estimator some individual MSEs get worse to yield a lower overall MSE. I was focusing on your example and assuming you meant this was only possible because some individual MSEs were larger than others to begin with (under the individually optimal estimator).