No, it's not okay. Today's electrical standards use a grounded plug (and outlet) and you should upgrade the cord and install ground braid as described in Vid's guide, post 15. https://pinside.com/pinball/forum/topic/replacing-line-cords-plugs-wall-sockets-vids-guide#post-1945124

Then when you're done turn all your machines on and check to see if any voltage is present when you touch the leads of your DMM to adjacent machines. If any is present fix it immediately.

Nick, can you post a picture of the inside of a bingo that you think would not be safer using a three prong grounded cord & ground braid?

You've raised some good points. Thanks for contributing to the topic.

Which basic electrical theory principle applies to his statement and how does that principle being mis-applied make his games less safe?