That's my understanding as well, FWIW. This is how I would phrase it:

Shrinking helps. In R^d there's no such thing as shrinking generally, only shrinking in the direction of some point. (The point that's the fixed point of the shrinking.) Regardless of what that point is, it's a good idea to shrink.

The James-Stein estimator does not respect translational symmetry. If I do a change of variables x2 = (x - offset), for an arbitrary offset, it gives me a different result! Whereas an estimator that just says I should guess that the mean is x, is unaffected by a change of coordinate system.

This is a big problem if the coordinate system itself is not intended to contain information about the location of the mean.

This makes sense if "zero" is physically meaningful, for example if negative values are not allowed in the problem domain (number of spectators at Wimbledon stadium, etc). Although in that case, my distribution probably shouldn't be Gaussian!