This is exactly why I don't ground my machines.
Breakers weren't designed to protect people from getting shocked. They protect from over amperage and a dead short but when you are between the path to ground, it is not a dead short. Your body provides resistance. The breaker doesn't trip
I have a lot of games if I tried to ground them all, sooner or later I would miss one or procrastinate grounding it, or miss a connection point somewhere . Then I would create the most dangerous condition, One grounded, one un-grounded.
The problem is people keep doing this to make their machines "safer". But in some situations they have actually made them less safe. The only serious shocks I've ever heard of were in situations just like the one JR described. An un-grounded machine beside a grounded machine. Which is more dangerous than if both machines were left un-grounded.
So ground them if you want, but make sure you ground every single machine in the room.
What I do, can best be describe as, sort of, double insulating. A large majority of my mostly early 70's machines do not have line voltage at the door. Those that do, I take extra precautions to insulate. I replace and add extra fish paper and I line the underside of the door switches with a thin piece of plastic to insulate everything from the metal. So a loose wire or bent switch can't come into contact with any metal. Also, I put those little rubber boots over the toggle switch on all my games, And I use GFCI outlets.
I think grounding is providing people with a false sense of security. I rather do things to avoid a shock in the first place than to rely on breaker to protect me, because it won't.