There are 2 moments where I know the game will always auto launch and that's if
1) Switch 56 "Ball shooter" is closed while ball is in play.
2) If a trough switch remains closed after feeding the ball shooter, it will feed again and automatically shoot the balls.
When the ball launches, do you hear the casting reel sound effect ? If the trough switches are closed and it kicks the balls you wont hear that sound effect. Also , the plunger will still be "charged" and pressing the cast button will make the reel sound effect , and activate the plunger even without a ball in the lane. Only when pressing cast does it make the sound effect.
You rebuilt this machine, did you change any switches ? Did you verify that the pinout of the switches you changed were the same as the original ones ?
This a crap shoot , but you've swapped all these boards .. Have you tried a different ribbon cable? It has to be something that remained behind during your tests.. The wiring , the switches, the ribbon cable.. I re-wired the 3 trough switches mixing the wires , but the game doesn't seem to care if trough 1 is wired as trough 2 and etc.. it only reacts if a ball is out but a switch remained closed. I had an issue in the past where jet bumpers would close when the others we struck.. turned out the vibrations alone from hitting one bumper was enough to close the leaf switch on other bumpers , causing them to close. I wonder if something similar might happen, where as it kicks out the ball, vibrations make another trough switch close and the game then launches it..