Well, I'd see if the reset line, pin 1, changes when you fire the auto launch. But to be honest, that would happen no matter what. Because either the reset line is getting bad feedback from that chip (it should just be reading that line, not outputting to it), or the voltage is dropping low enough to make the watchdog send a reset signal. The issue is that a whole bunch of chips touch the reset line. But only that chip touches the reset line and specifically deals with firing that solenoid. I just think it's the best choice. Here's my process of thoughts:
Okay, game resets when a coil is fired. But it's not that coil, because it does it when any coil is hooked to the line. No other coils on that power source trigger it. That means 99% it's not wiring and it's not the coil. That leaves the head. You've tested the transistors and they're fine. What is behind those in the schematic? Small caps and resistors. Unlikely to fail, and even more unlikely to fail in a way that causes resets. So what's behind that? U1, and it has direct access to the reset line. It's where I would look next. But there's no way to really tell. I guess you could cut pin 1, and see what happens, as you're gonna change it anyways. But it might kill the firing of all those solenoids. I don't know why it reads the reset line in the first place.