In the Megillah, Mordechai saves the king's life from Bigsan and Seresh, and by doing so, he ultimately gets paraded around the city by Haman.
Let's say that this subplot never happened. How would the story have been different? Esther still would have been queen. Haman still would have wanted to kill the Jews. Mordechai and Esther still would have had their fasts and their conversation. Esther still would have asked Achashveirosh and Haman to her parties, at the latter of which she still would have had Haman killed.
I see two differences between these two plotlines. First, if Mordechai wasn't in the king's book, then the parade wouldn't have happened, obviously. But more importantly, if Achashveirosh could sleep that night, not having an unsettled account rattling around in the back of his mind, Haman would not have been able to ask about the gallows in any event; and if Achashveirosh couldn't sleep, perhaps the officers would have been able to get him to sleep with the Royal Chronicles, which was their goal even as the story actually played out. So ultimately, while that night would have played out differently, the end result presumably would have been the same.
The second one is Charvonah's jumping in at the end of the second banquet. If Haman was trying to kill Mordechai, who had saved the king, what else would Haman do? So Achashveirosh killed Haman. But let's say that Mordechai never saved the king. Charvonah still might have accomplished something: after all, Mordechai was still an advisor to the King, as we see from 2:19:
וּמָרְדֳּכַ֖י יֹשֵׁ֥ב בְּשַֽׁעַר־הַמֶּֽלֶךְ
And Mordechai was sitting in the palace gates
as explained by Megillah 13a:
אזיל שקל עצה ממרדכי אמר אין אשה מתקנאה אלא בירך חבירתה ואפי' הכי לא גליא ליה
[Achashveirosh] went to get advice from Mordechai [on how to get Esther to tell her nationality]. [Mordechai] said: "Nothing makes a woman jealous like the thigh of her friend." But even so, she did not reveal to him.
Since Achashveirosh obviously held Mordechai in high regard anyway, Charvonah still may have had his intended effect.
So what part of the story would have been different had the Bigsan and Seresh subplot not happened? (That is to say, they never plotted to kill the king.)