Hi, there is another tack here we should try first. Earlier Data East machines (like the earlier Williams System 11 design they copied) had a number of what were known as "special solenoids". They were basically solenoids that required a really quick reaction, like pop bumpers and slingshots. The original thinking was that registering the switch being closed on the sling shot, getting that to the CPU, then having the CPU decide to fire the slingshot was going to be too slow. For those 6 special solenoids, the switch on the sling shot (or pop bumper, kicker, whatever) was connected to the driver board so that it could directly fire the solenoid. There would then be another switch on the component somewhere to tell the CPU it had been hit, so it can score points etc.
In these games it was also possible for the CPU to fire the solenoids too, if it wanted too, which meant that they could be used for other things. As time moved on, it was realised that the CPU was fast enough to handle these coils itself, and the technique of having the special solenoid switches take care of firing the solenoids directly was dropped.
However......
The circuitry for connecting the special solenoid switches and allowing the direct firing was left on the board. To allow the board to be used in older machines, ie keeping it backwards compatible. That apparently unused section can screw up, causing coils to lock on. At this point I shamelessly copy and paste from the old pinrepair site :
On games Back to the Future and later, there is a potential problem with this
backwards compatibility. The old special coil circuitry can be damaged, and
cause the special coils to lock on. This can be very frustrating. For example,
say a pop bumper coil locks on immediately when the game is powered on. After
replacing its associated TIP122 transistor and the 2N4401 pre-driver transistor,
and checking the coil's diode, the coil is still locked on! So go back a step
further, and replace the 7402 chip at location 12A or 12B. And still the coil is
locked on. Finally go all the way back, and replace the 6821 PIA chip at
location 8H or 9B. Yet still the coil is locked on!
The problem could be in the parallel, but unused, old special coil circuit!
If just one of the small capacitors at locations C40 to C45 (.1 mfd) shorts, its
associated coil can lock on. This is a very common, but confusing problem. Also,
one of the diodes at locations D1 to D6 (1N5234 or 1N4735, 6.2 volts zener), or
the resistor pack at R24/RA24 (4.7k x 8) can short (less likely, but still worth
checking). In CPU Revision 3 games, these components are unused, but they can
still become damaged!
On games prior to Back to the Future, I have seen a cases where a special
coil would not lock-on until a game was started or the game entered diagnostic
mode. This is a clear that the problem is *not* the driver transistors, but is
related to the Special Solenoid's switch circuit. In the case I repaired it was
one of the D1 to D6 1N5234 zener diodes next to connector CN18 (the special
solenoid switch input connector). These diodes are easy to test using a DMM set
to diode mode - .4 to .6 volts (or 1.8 volts if the DMM leads were reversed),
except for the shorted diode which tested 0 volts.
Now back to me
Tracking back through the SW schematic (which has some crap labelling), it looks like the cap and diode pair you should suspect are D5 and C44, and as mentioned above it could be the resistor array at RA24. The diode you can test per the notes above, also compare it to the other diodes next to it. C44 is tricky to test in place. The resistor array should show 4.7k between it's pin 1 (marked with dot normally) and each of the other 8 pins. In particular to pin number 8, which is the one connected to the redundant but trouble-making switch old switch line for the left slingshot.
Good luck. I'm sure we can get this worked out and fixed, if you have the patience

Mark