Any voltage/current leak finding it's way into the ground braid would not actually travel through the legs in order to find a path to ground, unless perhaps if the game is standing on a concrete or earthern floor. It would likely travel through the ground wire of the power cord, and into the receptacle, provided it is properly grounded. A GFCI outlet would trip and cut power to the game as well. You would be safe from a shock even if the legs were not grounded. The current wouldn't travel through your legs to get to ground, unless maybe if you were standing barefoot in a puddle!

I think you've answered your own question. I'm with you in thinking the legs on modern games are grounded just for regulatory compliance.
The worst shock I ever got from a pinball machine was when I touched the grounded leg of a game (reaching down for my beer), while having my other hand on the lockdown bar of an adjacent game which obviously had a hot lead in contact with a metal part of the game (likely the coin door). Pretty bad jolt, and if the leg of the other game wasn't grounded, I wouldn't have gotten zapped. Of course it alerted me to the fact that there was a problem!

Breaker? Where? It was in a friend's basement gameroom.
As far as I remember, nothing tripped.