forget the switch inputs for now. start at the coil and work backwards. I am assuming the driving transistors are Q70/Q71.
0) Game on, put DMM on either lug of coil. should show 35 volts DC.
1) Ground the non-banded coil lug with a jumper wire. coil should fire.
2) ground the transistor lug for this coil (q71) at the cpu board. coil should fire.
if steps 0 to 2 don't give results, you probably have a power issue.
Turn game off...
3) remove connect J19 from the CPU board and using DMM set to diode function (and game off), test the Q71 transistor. black DMM lead on the metal transistor tab. Both outside legs should show .4 to .6. If not, transistor probably bad, replace with a new TIP102. After done testing, replace connector J19.
If still not working....
4) Now turn game on and put the game into coil test, and select this coil. (use the middle diag switch to "stop" the test from moving. this will continually fire that coil, over and over.)
5) use the logic probe. Assuming the coil transistor is Q71, put the logic probe on u45 pin 5. It should "beep" about every second, as the PIA tries to fire the coil. That is, it should be HIGH and momentarily go LOW as the diagnostics pulse the special solenoid in question. If you have nothing here, the PIA chip is probably the issue.
6) if you have a signal at u45 pn 5, look at u45 pin 6. this is the blanking. this should be LOW all the time. Note when the game boots and is not in play or in test, blanking will be HIGH. When a game is started (or diagnostics are started), blanking goes LOW. Pobably not an issue, but worth mentioning. if blanking is high or missing, none of the other coils will fire, so it's probably not the issue (unless there's a broken trace.)
7) What the 7402 chips is doing is looking for two LOW inputs (the blanking, which is always low, and the PIA signal, which is high but goes low for just a moment as the desired coil is pulsed). Now go to the output pin u45 pin 4. the coil test is still running, and this pin should bounce HIGH every second or so (as the diagnostics pulse the coil in question). What is happening is the blanking (pin6) and the input (pin5) both need to be LOW for the output (pin4) to trigger HIGH. if you have no signal on u45 pin 4, the problem is the u45 chip. replace it (and use a socket).
At this point, you can follow the signal to Q70 if you want, and to Q71. But frankly unless the board is damaged (broken trace), the issue is probably Q71 (tip122 but replace with tip102). While you're at it replace q70 too (2n4401). Note the test in step3 is not definitive. Even though the transistor tested as "good" in step3, doesn't mean it is necessarily good. (Opposed to it testing bad, where it means that transistor is definitely bad.)
with this done, the coil should fire in coil test. If the coil fires in test, but not in the game, we can test the switch inputs. But 98% of the time, the switch input is not the problem.