Stern pinball's tech support is actually quite good with these kinds of questions, so I recommend you contacting them.
I've always worked under the assumption that the GAME eprom handles the game logic. It boots the game, and starts waiting for switch inputs. When the test switches are activated, it's the Game eprom that decides to go into Diagnostics, and to move around in the menus. If you select lamp or coil test it's the Game eprom's program that tells the IO board to apply voltages to the lamps or coils.
The display controller board has an eprom at U5,U6 that holds graphics and animations, but it doesn't control anything except for the dot matrix display. I'm working under the presumption that the CPU sends a command like 'Start Animation 01' and the display controller will figure out where that animation is in it's eprom and do what it does. Similarly, I suspect that the CPU says 'Text display 'THIS IS A MESSAGE'' kinds of signals get turned into the correct kinds of dot matrix alphanumeric characters at the display board to display THIS IS A MESSAGE. The display controller board has no connection to the input switches, so it is unlikely that it is controlling the CPU's switch inputs and IO board outputs.
Similarly, the Sound Eproms I suspect have a table of various locations where the samples and note lists are stored in memory, and a small amount of program code that the sound processor uses to allow the CPU to send it commands like 'Play Song 01'.
In terms of troubleshooting, I've had bad Display Eproms, and bad ribbon cables. I've had a battery acid destroyed Display Controller board.
In general, the Display stuff on the SAM system Stern pinballs has just worked for me. I've never had to go too deep.
Let us know what your actual diagnostic challenge is, and maybe we can help you. Definitely contact Stern technical support when things aren't making sense.