Question: I understand that it was considered a good "simmon" if the goral lashem came up in the right hand. I do not understand what would be accomplished though by switching the goral from the left to the right hand (in the question to Rabbi Akiva) I assume that if the goral came in the left hand it might cause the kahal to do teshuva so switching it might defeat that purpose
You have raised a very interesting question. First, I would like to clarify one point. Although the "Lashem" coming up in the Kohen Gadol's right hand is a good sign, the "Lashem" coming up in his left hand isn't necessarily a bad sign. It does not show that Klal Yisrael were not forgiven, nor does it show that they were forgiven. It just does not show either way. If so, the students of Rebbi Akiva wanted to know if, in order to improve the public morale, the Kohen Gadol could switch the "Lashem" to his right hand.
The SI'ACH YITZCHAK inserts your question into the Gemara. He learns that the students of Rebbi Akiva wanted to know if the "Lashem" lot could be switched to the right hand in order to honor the Kohen Gadol, or if it was better to leave it as is, since "La'Azazel" appearing in his right hand might cause the people to repent.