I had seen the ‘preferred’ behaviour in the Slice Controllers of the View Controllers module before — although I’d temporarily forgotten about it, so it’s good to be reminded.
However, it seems that in there the volume would have to be specified three separate times — once for each orthogonal slice.
So it seems that the feature I was talking about is already present!
I notice that it is necessary to right-click on the eye for one of the volumes in the hierarchy (it won’t work when clicking on the eye for a segmentation, for example), and thereafter the setting is automatically applied to all other volumes open in that instance of Slicer.
It appears that the preference is also preserved for new instances of Slicer.
P.S. It never occurred to me to right-click any of the eyes in the hierarchy.