Doesn't really make sense to replace stuff before doing some basic and easy troubleshooting. For several reasons. Not the least of which, you might introduce a problem with something that wasn't broken in the first place, compounding your problem. Just put a voltmeter across the wires that would be going to the coil and turn the game on. If you see voltage then you know that the coil would have locked on, if it were connected. If you don't see any voltage then run the solenoid test and see if you get a voltage spike on your meter when that solenoid should be energizing in test. It can be hard to see with a digital meter but should still be obvious. An analog meter will be more obvious.
Alternatively you could do a continuity check between the metal tab of the transistor driving that coil and ground but I wouldn't consider that as reliable a test as using a voltmeter with the game turned on.
If you determine that the coil is locking on then a shorted driver transistor is the most likely cause but obviously not the only possible cause. I still wouldn't just go replacing stuff. I would probe through the pre-driver with the game on to see if anything before the driver transistor is locking on but that would require the schematic and some idea of what's going on in the circuit. An easy approach would be to desolder and test the driver transistor, as it would be the most likely cause of a coil locking on. Keep in mind that the driver transistors on Gottlieb System 3 are MOSFET (IIRC).
Are the transistors on the PCB's in the back box? Nothing looked shorted really..
You're not necessarily going to see evidence that there's a problem. Many times smoke will pour out of the driver transistor when they short but not necessarily. It's as likely that it blew the solenoid fuse and someone figured out that the coil was locking on and disconnected it. Of course, there's no reason to assume anything. Just do the troubleshooting.
How would I go about identifying which transistors and where they are located? I could just unsolder them and test with my DMM for shorts, no?
Normally I would point to the manual but you won't easily find a Stargate manual online (Gottlieb). You should be able to determine which transistor it is with a continuity check between the ground wire at the coil and the metal tabs of the driver transistors.
As far as connecting the coil goes, the positive side of the supply voltage connects to the banded side of the diode. That means it will NOT be conducting when the coil is energized.
If you have 3 wires going to the coil, more than likely one will be the ground side coming off the driver transistor and the other two will be the positive side coming right off the power supply. The positive side of the coils are daisy chained together. One of the two wires will be supply from the last coil and the other will be going to the next coil in the daisy chain. The daisy chained wires should be the same color, generally speaking.
I actually noticed the solenoid wasn't even tightened down to the wood so it does look like it was replaced or at least taken out at some point in time.
It's possible that someone removed the solenoid assembly for some other reason and cut the wires to do it. Like cleaning, broken linkage, etc... That's why it makes sense to troubleshoot the problem before replacing any parts.