The obstruction in the coil is a melted sleeve, which means it locked on solid. Unless you find some other kind of physical short (frayed wire, or a loose, shorted bracket, etc) I would look at the corresponding transistors on the board. Typically there's no reason a ball kicker should short or stay on that way, so this is a pretty good indicator there are problems upstream.
It sounds like you're still learning the ins-and-whats of troubleshooting these things. Make sure you have the manual and schematic. But even that can only tell you so much and there is a learning curve to understanding it all. So, meanwhile, check every wire in the game, especially the playfield: gently poke, prod, tug for anything loose. Get familiar with what the majority of solder connections look like, so that you can spot any that look different, indicating a previous repair. Compare those to the schematic to make sure the repair was proper.
Obviously you know to post here when you get stuck...
Sometimes weird things just happen. When I was about 12 or so I was teaching myself wiring and repairs on HO scale model trains. All these years later I don't remember what I did in this particular incident, but I remember I was trying to create a signal block, had wired that into an electrically controlled track turnout with junk signal I was trying to get to work, and ALSO was testing a locomotive I'd repaired. I was doing all of this with NO instructions, just experimenting. Well I fired up the power pack and all the lights in the house went off - yep, blew the breaker just like your TZ. That "should" not have been possible, but needless to say after the "oh shit" moment passed I thought it was kinda awesome. I remember inspecting my work and nothing seemed recklessly out of sorts except I internally shorted the loco because I didn't realize it had another feeder circuit... but you can do that all day long on purpose and the only thing that happens is a loco that doesn't move, not a full-structure panel breaker reset. Never happened again in the subsequent 25+ years, but I still fondly recall that incident because who doesn't like some occasional "WTF stories" with their hobby?
You're doing the right thing exercising caution. Unplug the game, look through it methodically, and you'll get it figured out soon enough.