Restoring an old fridge; should I add a ground wire?
I don't see the harm in adding the ground wire.  All neutral wires end up going to the neutral bar in the main panel that is mounted directly on the metal box just like ground wires are mounted directly a ground bar that is mounted to the box. Sub-panels are different having an isolated neutral bar but everything goes back to the main. House panels here in California have the neutrals and grounds going to the same bar.  In all actuality if you go from hot to neutral with a volt meter you will get 120v and when you go from hot to ground you will get 120v. Adding the ground to the frame of the refrigerator will just be a more direct route for the ground to take at the fridge.  You might want to ohm it out between the hot and ground before testing on a life circuit, making sure there is no continuity.