The pause was not intended to aid the new game, but to give the previous game time to finish. Some engines resign after printing a move, so that their opponent has already started thinking when WB receives the resign, and terminates the game. Now some engines are unresponsive while thining, so there would be no way to stop the thinking engine before it has printed its move. And if you immediately start the next game, that move might come within it, where it is illegal. With engines that support ping this cannot happen anyway.
So it doesn't matter whether the pause is before or after the 'new', as long as it is before the 'go'. Because engines only start sending moves after receiving 'go', any move that comes in before it must be a left-over move from the previous game.