Daf Yomi - Yoma 40 - Switching the Lots

The students asked Rebbe Akiva if the lot For the korbon laHashem camo out in the kohen gadol's left hand, can he return it to his right. This question was based on it being a good omen for the goat which will be used for the chatos to emerge in his right hand. Rebbe Akiva answered that we should not leave room for the tzidukkim to ridicule us. The Gemora asks on the question, how could one possibly switch according to the Tanna that holds that the lottery is essential to the service.

Tosfos HaRosh and Gevuros Ari ask that this question should be universally accepted, fr even the Tanna who holds that the lottery is not essential would agree that if it was done, the lots have established which goat is for which korban, so how can the kohen gadol possibly switch it? The Mikdash Dovid (24 -3) answers that it is evident from here that according to the Tanna that holds that the lottery is not necessary, even if it was done, that was not what designated each goat to its particular destiny, rather it was the words of the kohen. Even if the lots indicated one way, the kohen (if not for the tzidukkim problem) could have switched them.

Another question can be asked. What would the benefit be to switch the left to the right? The result of the lots already shattered our hopes of the good omen?