So why is this fairly significant function tucked away in this relatively obscure window? Shouldn't there be a Zero Rotations feature as part of the Cloth Room itself, or as an item in the menu bar? (I'm aware of the "Zero Figure" and "Restore Figure" menu items, but each of them have slightly different behaviors.)

There should be a simple "Zero Joint Rotations" button on any Object panel for a Figure in the Pose pane. There should... but there isn't. It's such a basic  function, and so necessary, that I can't understand why it has been overlooked through all of Poser's iterations. A Zero Rotations for All Figures in a scene might be handy, too.
I'd also like the tool to work based on the parent/child relationship of the rig. So, if I selected the left arm and chose to "Zero rotations", any children would also get zero'd. If I selected the left hand, the hand and all the fingers would get their rotations set to zero, but the rest of the figure would remain posed as it is. If I selected the "Body" of the figure, everything would get reset.

They'll probably include something like that for Poser '39...

As far as what I use the Joint Editor tool for, aside for creating/tweaking custom content and third-party content, I use it to reset the object center's for props that must be rotated around a different center. For instance, for doors and windows, you can't just load up the prop and expect them to rotate around their far lef/right/whatever edges to simulate a door or window opening. So, open the joint editor and you can do this easily using the UI and dragging the object center to the desired location. Using a fixed camera position (left/right/top) will help you get it aligned properly without having to do any messy figurin' out how to enter the values in the axis boxes.

PS - Yes, I use it to zero figure rotations, too.