I figured it out. The Ball Release coil is fired by Q15, which is driven by Q14, which is driven by pin 8 of IC1. IC1 is a 7408 AND gate, which uses pins 9 and 10 as input. Pin 9 is the blanking signal and pin 10 is the signal from the PIA. If blanking is high (which it should always be after initial boot-up) then all it takes is a quick pulse from the PIA to pin 10, and that puts pin 8 high long enough to fire the coil.
I checked pins 9 and 10 and found that pin 9 was always low, despite other blanking signal pins on the same chip being high. That's odd. If that were the case, then why would the coil fire at all? When the PIA sends the pulse to pin 10, the low on pin 9 should cancel it out. Clearly that's not happening. I remembered reading about how TTL chips can "float high" - meaning that if an input pin has a non-standard amount of voltage, the chip assumes it's a high signal. This non-standard amount must be messing with the chip, and my logic probe. So what's causing the weird voltage on that pin?
This.
IMG_2501.JPG
Pin 9 is the second from the right on the bottom row - the one missing a solder pad. What's funny looking as well is that second from the left on the bottom row is also a blanking signal input pin, as is the second from the left on the top row. I have a feeling those two weren't joined like that when the board came off the factory line. I'm guessing that someone hacked the board to get IC1 to work properly, but they didn't finish their hack, and left pin 9 in the lurch.
I soldered some 30-gauge wire wrap between pin 9 and pin 13 and tested the board on the bench with a Siegecraft solenoid tester module. The coil no longer fired on power-up. I put the board back in the machine, and it was the same thing there. Another problem solved.
Thanks, everyone!