Put your meter in continuity mode. (Would help if your meter beeps when the circuit is made.)
Put one lead on the DRIVER wire (the THIN wire, not the THICK, 50v wire!) and the other multimeter lead on your smetal siderail.
Go into Solenoid test, and tell the game to fire that coil.

If you meter beeps every time the coil should fire, then the transistor is working.
If the meter does not beep, then the driver wire is not being brought to ground.

If the latter, then do a continuity test between the driver wire on the coil, and the driver wire on the PDB. If you do NOT get continuity, then the wire is broken - find the break and fix. If you DO get continuity, then the driver circuit (driver, pre-driver, etc) is bad on the PDB.

While true, this wouldn't be the case with his issue - he says the rocket is not firing. If this transistor was shorted on, that would mean that the rocket kicker coil would be locked ON, and overheating slash smoking.

True. I'm not sure if in this case it would be a useful test to try grounding the transistor tab with the game on, and see if it fires. I know that's a way to test wire continuity, but NOT sure if that actually tests the *transistor*. I'm thinking no.

Yeah, Swainer80 mentions to fire ALL of them on that row in coil test.
THis is how you find out where the coil power feed is broken. i.e. If the gumball Popper fires, and another one doesn't, you know the break is somewhere related to the GUmball Popper.

*Note - the order of those coils in that 'row' is NOT the order that the +50v coil feed follows.

Edit also: And yes, if you do not have +50 on the violet wire, then... while you CAN check the transistor, it's a meaningless test as the problem isn't on the driver side.