The lynching bee fails, in my opinion, because it is not organized by real feeling but instead by whimsy. The men in the mob knew that Sherburn would shoot the drunk if he crossed the line. They could have stopped the killing. But they didn't because they didn't care enough and/or felt that Sherburn was within his rights to shoot the man after fair warning. When the men collect as a mob later, they are adopting a new view of events, one which they did not adhere to at first.
This maleability in the mob suggests a superficiality of feeling.
This scenario is part of Mark Twain's tableau of the various types that compose human nature. The man of integrity, albeit wielding a shotgun, is by sheer force of character able to intimidate. For the weak of heart, fortitude is often a formidable adversary.
Another American writer, Harper Lee, depicts Atticus Finch facing a mob in a simlar scenario.
In the face of Sherburn's unflinching courage and under the shadow of his scorn, the mob had no choice but to quail in their own cowardice, turn, and run away. The fact that he held a shotgun helped too.
In Chapter 22 of Twain's Huck Finn, Sherburn is correct about the mob. He tells the members of the mob that they are not real men, but cowards. He criticizes them because they have enough "pluck" to tar and feather "poor, friendless cast-out women...", but that none of them is man enough to carry off a lynching. He tells them that they borrowed their courage from "half a man," Buck Harkness. In essence, it seems that Sherburn is the only man assembled among the group. The mob seems to see the wisdom in what he says, for they all run away, including Buck Harkness.
Sherburn is correct in his assessment of the mob. They far outnumber him as an individual, yet are cowed by the stand that one man takes. Even though Serburn was armed, they could easily have overpowered him with minimal casualties. They are gutless scum precisely because they are whipped into a frenzy but have no courage and strength to pursue their objective whatever the cost.
The "Lynching Bee" failed because Sherburn showed himself to be one who would not be swayed by the mob. Even when faced by the whole group, they didn't have the courage to confront one determined individual who intended to do whatever was necessary to stop them. Their conduct confirms Sherburn's opinion of them as being cowards.
Sherburn thinks that the men in the mob are basically gutless scum. Just look at the way he talks to them. You can easily see that he has no respect for them at all.
Is he right? I think he is. After all, these are people who need to get themselves all worked up (and in a mob) before they can confront him at all. After he calls them cowards they sort of all just slink away. It's hard to argue that he is wrong given how they react.