the line unit can't reset if the coil wire is broken :-)
refer to post #5 wrt taking off the paper wrapper and seeing if you can find the wire end that is under the paper. If I had to guess, you're pooched and will need a new coil. You'll need a soldering iron to install it, so if you don't have one it's time to buy tools.
Something like this will work fine and comes with solder:
https://www.amazon.com/Soldering-soldering-solder-adjustable-temperature/dp/B09DY7CCW5there's also internal battery, usb-powered and units that use power tool batteries (like dewalt, milwaukee, etc) that are overkill if you rarely solder, but kinda handy if you work away from places to plug into. They cost more tho, and make sure to read reviews. You care about "recovery time" - time to reheat after soldering something - and battery life.
if you're reading this and have never soldered something, watch a couple youtube videos on the right way(s) to solder. Putting the solder on the hot iron and dripping it onto the wire is not the right way. If you want the foolproof method, heat the wire and apply some fresh solder to it so it obviously melts into the entire wire. Heat the lug and apply fresh solder to that. Then hold the wire to the lug and heat both until the solder is obviously molten on both pieces and remove the iron. Hold the wire a few seconds for the solder to solidify. You may want to use gloves or pliers to hold the wire, but if pliers don't hold so tight you damage the plastic insulation when it's hot.
the coil wobbling around is normal. As long as the coil stop (the end piece with the two screws) is tight, the coil can't go anywhere.
it's best when manually stepping the unit to push the plungers into the coil so you get the right amount of travel on the arms/pawls. The unit at reset position will light the 1st coin, then for each step up it moves the lights. The unit therefore can step up 4 times to the flat spot in the ratchet where it can't physically step up more.
to test reset ratchet movement, manually reset and step up the unit once, then manually reset again. You can look at the wiper side and see if the wipers contacts have moved more-or-less completely onto the right rivets. There's a bit of slop/misalignment at the reset position, but at each step up, the wiper contacts should be pretty centered on rivets. If the wiper contacts aren't centered well enough at step 1, you adjust by loosening the two screws that fasten the contact plate to the unit frame and twist the contact plate a little. Do not do that adjust at the unit reset position tho or you may step up the unit and find the contacts are way off on steps 1+.
the problem with resetting from step 1 is you don't have a lot of momentum and the white peg in the ratchet needs to push the open-at-zero switch blades. If the blades have too much tension (bent too much toward the mating blade), the ratchet rotation can stall. Not enough blade tension and the switches may be unreliable.
It's a balancing act and david's suggestion of relocating the center torsion spring end to the next hole clockwise will likely fix it (i.e. move the end from the 9:00 hole to the 12:00 hole) ... easiest if you unhook the long end from the peg. I'd remove the entire torsion spring and clean it, then reinstall with either the current 1-3/4 turns or go to 2 turns if you need more reset force.
after repositioning the torsion spring, verify you can manually step the unit all the way up (4 steps).
the unit should work with 1-3/4 turn of tension, but if things are a bit cruddy it will slow down. Adding tension to the torsion spring is the quick fix, but you can't add so much that the unit can't step up to the top position. This is more a problem on the payout counter since those wipers spin 720 degrees and if the torsion spring is too tight, the rotation will stop before payout ends and you get runaway payout until the machine shuts down (or the hopper empties).
if you want to play the game without the line unit resetting, stick paper between the switch contacts on the open-at-zero switches (see blue arrows in pic in post #5). You mainly care about the switch with the yellow and grey/white wires. You want that open so the line unit reset relay coil doesn't stay powered.
note the line unit reset relay/coil is different from the line unit reset solenoid coil that is kaput. On your game, the relay is probably an "ice cube" style one.