OK, pulling out the 555 gives the machine a chance to enter one of two states. I'll refer to them as "broken" and "working".
The "working" state has the appearance of just that - everything's working. The +5 and Blanking LEDs are solid, and the diagnostic light is flashing. Everything works. Before I stripped the machine, I was able to play a complete game and everything was peachy.
The "broken" state is one where the diagnostic light never flashes, the bell solenoid locks on and the +5 and Blanking LED are solid. (There's also a "Broken State B" that I've only encountered once, which has a minor variation. I'll cover those variations at the end.)
I checked all the empty pin spots when the machine is in the broken state.
Pin 1 (Ground) has zero voltage and read low on the logic probe. This is presumably because it goes straight to ground.
Pin 2 (Trigger) has 3.3 volts and reads high.
Pin 3 (Output) has 4.5 volts and reads high.
Pin 4 (Reset) has 4.9 volts and reads high.
Pin 5 (Control) has zero volts and doesn't read at all on the logic probe. This could be because it goes through a capacitor first, according to the schematic.
Pin 6 (Threshold) has 3.8 volts and reads high.
Pin 7 (Discharge) has 4.5 volts and reads high.
Pin 8 (Power In) has 4.9 volts and reads high.
Here's the info for when the machine is in the working state.
Pin 1 (Ground) has zero voltage and read low on the logic probe. This is presumably because it goes straight to ground.
Pin 2 (Trigger) has 3.3 volts and reads as pulsing.
Pin 3 (Output) has 4.5 volts and reads high.
Pin 4 (Reset) has 4.9 volts and reads high.
Pin 5 (Control) has zero volts and doesn't read at all on the logic probe. This could be because it goes through a capacitor first, according to the schematic.
Pin 6 (Threshold) has 3.8 volts and reads as pulsing.
Pin 7 (Discharge) has 4.5 volts and reads high.
Pin 8 (Power In) has 4.9 volts and reads high.
From the broken state, grounding pin 40 on U15 will reset the machine and put it into the "working" state. Grounding pin 4 on U43 (the spot for the chip in question) will have the same effect. This is the first confirmation I have of spot U43 actually having some kind of positive effect on the game.
I think I understand why grounding pin 4 of the empty spot has the same effect as grounding ping 40 of U15 (as well as U10 and presumably others). If I follow the schematic, I see that the two are connected, so grounding one also grounds the other. But based on what I know the basic usage of the 555 timer chip is, I don't understand why the reset pin is the one that controls the grounding of other chips' reset pins.
The only thing I can think of is that it's not so much as the grounding of pin 4 that drives the reset of the other chips - it's that pin 4 is part of a larger circuit that gets grounded at some point along with all the others. What is the condition under which all these joined pins would get grounded at the same time?
And with that figured out, I move back to the point of the 555 timer within the scope of the whole system. If I understand how a 555 timer is most commonly used, it's designed to keep the output pin in a certain state until it detects a change in voltage of an attached capacitor. Depending on how its wired, it can either function as a "one-shot" pulse generator, or be used like a flasher circuit, alternating pin 3 between two states. There's some kind of third use that I still don't understand, as well.
How does a System 11 make use of the timer chip? Perhaps that will help me figure out why putting one into the board starts making things act weird.
(NOTE: There's also a second broken state where all three LEDs are solid. The voltages are the same as the working and primary broken state, but pins 2 and 6 are pulsing rather than high, just like the working state. I've only seen this state once, so it might have been a fluke.)