So we're getting to the limits of my experience here, so take this with a grain of salt, but:
The CPU gets an interrupt signal every 1ms on pin 4, which triggers it to strobe the displays, lights, etc. At that point it selects the IC18 PIA by turning its CS1/2, E, RW, etc inputs on, then sends it some data via D0-7 which should be immediately displayed on A0-7, including pin 4. Since you say that the outputs of the PIA work with the test ROM, it's probably safe to assume that, if those conditions were met, the PIA would be strobing. You've also said that all those inputs are strobing, so we know there's no failure in the bus drivers. Which means that, although the signals are getting there, they must not all be getting there in the right combination to actually enable the PIA, which means the CPU isn't sending data to the PIA (just other chips which look for a different combination of the same inputs the PIA is watching). Since the CPU is getting its trigger, that must mean either the CPU isn't watching for the trigger, or something internal isn't getting to the strobe logic.
This, however, I'm not sure is a valid assumption. Sadly there's no way to tell that the game ROM is actually running (well, besides the blanking signal from PIA pin 4, but that's our problem!), just that the game ROM has started at all (when it turns the lights off). The diagnostic button works by triggering an interrupt, it's not polled, so even if the game ROM is stuck somewhere and has stopped running, the button will force the CPU to jump to some other code temporarily to run the test.
Leon's rom is very simple, and doesn't actually test a lot of the circuitry. All it tests is the IO PIAs and the memory. It doesn't pay attention to the 1ms interrupt signal either, and doesn't use the other ROM sockets at all.
My first instinct would be that one of the other rom sockets, or the circuitry connecting them, is bad. I'm not seeing in a quick scan through the thread if you've specifically replaced all the used rom sockets, and checked for proper activity on them. You've replaced the CPU and PIA sockets, but have you done any others?
I guess it also might be possible that something on the driver board is crashing the game ROM, would it be possible to try a different driver board just to be sure?