Nice work figuring out what the problem is! This is super common on WPC machines from the era of your Demo Man. As others have said, the fix should be pretty simple by replacing that connector with a better style Molex. Also, while you have it unplugged you should wiggle the pins in the board to see if they are still solid. It is possible that some of them have developed "cold" solder joints on the underside of the board which is basically a cracked solder joint that is intermittant. The only way to fix this is to unplug all of the connectors, pull the board out, and reheat each solder joint while adding a little solder to reflow the connection. That's getting next level if you haven't done much soldering before, and really only needs to be done if you still have problems after replacing the plug.

There is plenty of info on how to do this on Pinside and elsewhere, I always check Pinwiki.com and Pinrepair.com. Go check out http://www.pinrepair.com/connect/ and read up on what you need to do and how. Here are the pieces and parts you'll need at https://www.marcospecialties.com , adjust quantities as needed and always have a few spares! You'll also need a wire stipper (not shown).

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The connectors currently in the game are insulation displacement connectors (IDC). The kind you want to replace them with are the crimp & insert, preferably the Trifurcon style.

After you get the parts, take your new connector and first insert the blank key filler so you know which ends are pin 1 and pin 11. Then starting at one end, cut the wire off the old connector, strip it, crimp it, and insert it into the new connector. Don't try and pull the wire out of the old one because there will be broken strands of wire inside. Just cut it off as close to the connector as you can where the insulation isn't discolored and crispy.