I don't have a RF schematic, but the outhole kicker ('Ball Return') coil is usually activated by a switch on the Ball Return relay 'O' along with a score motor switch (like at position 4C), and one or two other switches. One of them is often on the 'Last Ball' relay.
Since it's working at game start, and there's a definite behavior, I'd suggest observing if the 'O' relay is activating at that point (it should close, lock on, and stay closed until the outhole kicker fires), and then debug from there. (If 'O' is not activating, that points to the path feeding 'O'; if it is activating, then that points to the Ball Return logic, and possibly the 'Last Ball' logic).
The schematic will show the details of these two paths.