Since all the flippers stop working, I wonder if the Flipper Enable Relay is de-energizing. It should energize and stay energized as long as a game is going on. If it drops out, it shuts off all the flippers. The IC at U4 and transistor Q15 should be holding it on.
Could the diode on the coil of the left flipper be backwards? Maybe after the flip, the coil voltage collapses and goes back up the line to screw with the MPU and make the Enable relay release?
Just a guess. There is some useful information about Bally/Stern flipper configurations here: http://stevekulpa.net/pinball/bally_flipper1.htm