Being a multiplayer game, it probably only has one ball so the ball release coil should stay energized until the game is over. There is probably a switch in the ball trough that closes along with a switch on the ball count unit that together indicates all balls have been played and then releases that solenoid thru a relay when the game is over.

According to the schematic I have for Whirl-Wind (which is only slightly newer), the ball release coil stays energized from startup thru a switch on the last ball relay that is normally closed until the last ball relay is energized thru the balls played unit once you get to the last ball.

You say yours does not show it switches thru the last ball relay so maybe the circuits are slightly different, but operation should be close to the same.

Either way, that ball release coil should stay energized until the last ball or end of game.