In his story, Ambrose Pierce DOES use omnisicient narrator; he just does not use it throughout the story. In the different sections, there are variations in point of view: omniscient, in which the narrator knows everthing about all characters or events; objective, in which the narrator reports without comment, much as a camera would record a scene; and third-person limited, in which the narrator zooms in on the toughts and feelings of a single character. There is NO first-person point of view. [ First person point of view always has the narrative told in the first person. e.g. I noticed that the guard was getting the rope tied.]
In the first section, the objective point of view is used giving the reader only the facts of what occurs. That is, until paragraph 4-7 where the point of view switches to omniscient because the reader learns the prisoner's thoughts. And, Section I ends this way, so the reader does not clearly know what has actually happened as "The sergeant stepped away."
Then, Section II is told in the third-person limited to give the reader the feelings and reasons behind Farquhar's being on the bridge. Section III continues in this third-person limited, so the reader only knows what Farquhar is experiencing, making his dreamlike state seem real so that the reader is not convinced that Farquhar is dead. It is only the last paragraph that shakes the reader from the dream with the surprise ending in objective narrator. The reader is jolted, not just by the words "Peyton Farquhar was dead," but also by the shifting back to the point of view from which the story began.
Bierce's clever manipulation of points of view serves to provide that "suspension of disbelief" through the "something uncanny" of the Second Section that the reader, nevertheless, accepts. And, it is only through this very manipulation that Bierce's narrative is so effective.
In general, I think that the story would be a great deal less suspenseful.
In the story, Farquhar is hanged from the bridge. We then see the rope break and Farquhar falling into the river. At that point, we switch over to seeing the story from the first person point of view.
If the narrator at this point were an omniscient narrator, we would not have the sort of suspense and uncertainty that we actually have in the story. If the narrator were omniscient, we would know that Farquhar was dead the whole time. ???