I've been working my way through a United Pacer shuffle alley, and though I haven't gotten it fixed I might be able to get you pointed in the right direction. Someone more seasoned than I am might have better feedback, but since it's been a while I figured I'd weigh in.
First, check out Clay's bowler site. It's an awesome primer on this, as is most of the EM pinball documentation. http://www.pinrepair.com/bowl/
Most importantly, read the Before Turning It On section. Basically, it's not worth troubleshooting the electrical problems until you've made sure all the mechs are freed up and working properly. That's a lot of cleaning, but it's totally worth it. Get 2000 q tips, a gallon of rubbing alcohol, and some super lube and go nuts. Get a friend to help, and watch it cleaning the score reels.
After it's clean, you should make every effort to get the schematic for that game.
If it's Team Shuffle Alley, the service manual can be downloaded from: https://www.arcade-museum.com/manuals-arcade/United%201954%20TEAM%20SHUFFLE%20%20Service%20Manual.pdf. Even if that's not your game, it does a decent job of generally describing the operation of any of the United EM bowlers. You still need the schematic, though.
There ought to be some bulky connectors in the back of the machine that connect to the pin deck, lanes, and front coin box lights and switches. I'd replace those fuses, pull those connectors, then turn it on again. If the fuses blow, you know your most immediate problem isn't related to anything outside of the backbox.
If the fuses don't trip, turn it off and inspect each of the wiring harnesses for shorts, then connect them one at a time between power off/power on to see if the fuses blow.
You may want to to make a breaker fuse so you don't have to keep replacing fuses. You can buy a 7A breaker (they trip slower than a fuse, so you want lower amperage to stay safe) and solder it to a blown fuse. Then you put that in and just reset it when the fuse blows. This breaker should work: amazon.com link »
There's a million other things but really it just comes down to isolating the good sections from the bad. Someone with more knowledge of that specific machine might be able to help.
Oh - and that red wire up front - while it's possible that it's a problem, it's likely that someone rigged the game for free play and did some "tweaking" up there to make sure it stayed rigged.
Hope that helps...