If it below the wood surface, but is not out past the wood, the dip beneath the wood should make it easier to drill out:
I would just suggest heavy gloves maybe leather if you have them around, starting slow, maybe trying to give the top a little pretap, hold the front of the drill/the bit with one hand. I don't do alot of detailed carpentry, but I can see how that might be a little rugged for a pinball table, but if you are careful and take your time, I think it could come out ok, might depend alot on if it broke off flat or slanted, it if is easy to start the hole, the rest should just be being patient and keeping it straight, keep it straight.
Where is this on the table?