The proper way is connect the power supply neutral and ground directly to the incoming leads on the power cord. Connect the hot lead to the outgoing connector on the power switch.

This will result in your new power supply load being added to the existing load on the main fuse. You could install a separate fuse between the switch and the new power supply (sized appropriately) so a short might blow only the new power supply circuit rather than the entire game power.

Sounds like your hot wire is connected improperly from the wall, maybe reversed with the neutral. The surge protector should be interrupting the hot wire, making it impossible to get current from it while turned off.
Get one of these or something similar and test your wall outlets. https://www.amazon.com/Sperry-Instruments-GFI6302-Receptacle-Professional/dp/B000RUL2UU/

A faulty ground plug cannot cause the issues you described. If you have not confirmed your wall outlets are wired correctly you should do so.

You're getting current where it doesn't belong. A properly grounded game will dissipate that current safely but doesn't address the source of the problem.

Define "a little". As long as its not going to interfere with anything or lay somewhere to rub through the insulation and eventually short out it's fine.
If you're trying to store a few thousand feet for your next 457 projects no.

Don't assume. Read the post and do as it says. Here is the link again: https://pinside.com/pinball/forum/topic/replacing-line-cords-plugs-wall-sockets-vids-guide#post-1945124

Wood is generally a poor conductor so wooden items are generally not grounded. If you have metal things that could easily come in contact with something that carries current consider connecting ground braid to that thing.

Was your meter set on AC or DC? Have you compared readings with another meter to rule out equipment failure?

The ground wire/braid should touch anything that may have unwanted current present. Its purpose is to safely dissipate that current by providing a better path to ground than someone touching that thing would provide, thereby preventing the person from getting a shock.

It seems odd that a common failure of transformers is start providing current to the mounting lug. I think breakers would trip often with old games if that were the case.

Should be 4amp, not 16.
See page 96 in your manual.

WWtransformerjumperchart (resized).png

Connecting them in parallel is fine.
https://components101.com/articles/metal-oxide-varistor-mov-overview
"The MOVs should always be connected in parallel for increased energy handling capability and if you want to get higher voltage rating you should connect it in series."

Generally you'll solder the bare wires on the end of the cord to the proper place in the game.
There are many, many possible things that could have been done to your game since it left the factory so don't just go by 'standard' advice. Take pictures of what you have. If your cord is missing, say so.

Check the key posts in this thread and ask specific questions where your situation isn't covered.

Vid, I was with my new home inspector today and watched him test a GFCI outlet that was ungrounded. It didn’t trip. I’m assuming the button on his tester made a connection between the hot and ground.

Was it just an old GFCI that doesn’t function like newer ones? This house I’m looking at has no ground wire at the outlets (built in 1959). I presume the 3 prong outlets and GFCI were added later.

Will be interesting to hear my electrician’s quote for fixing that issue.

Three prong cords aren't typically polarized because it's impossible to plug them in wrong. To reverse the hot and neutral the ground prong has to be removed.

The voltage is still there and you still don't know where it's coming from but at least it's being safely routed to ground. That's a step in the right direction.