Why did Faulkner not write "A Rose for Emily" in order?
In order to best answer your question, let's switch and suppose that William Faulkner had written "A Rose for Emily" in order. Although the plot would have followed the same events, but merely in a straight sequence, an important part of the narrator's voice would have been lost in the process. In a typical third person narrative, we know what happens but we do not get the insight that would be obtained from a narrator who is taking the perspective of a first person who has seen and heard what went on in Jefferson. This is what we can appreciate in "A Rose for Emily".
Faulkner wanted to tell his story from the perspective of the townsfolk: Different voices who have seen different things at different times, all coming together into one narrative. It is exactly the manner in which you get information from neighbors when a huge event, or even a tragedy, happens in your vicinity: One person witnesses this, another witnesses that, and all the sights, sounds, and stamps that each person brings to the story are put together to somehow foreshadow what may happen next.
If you notice, the entire time that the narrator speaks, he or she uses the pronoun "We". These combined voices have witnessed Emily since her youth under the willful protection of her father, then as a grieving daughter, then as a spinster, and then as the rebellious older lady to defies all conventions and goes about with a Yankee of dubious origins. Suddenly, they see her no more, and all wonder what happens to Emily and to Homer.
Hence, all of these small but powerful details would have been hard to infuse in a third person detached narrative that lacks the spice and flavor of the "neighborhood gossip" that permeate the narrative of "A Rose for Emily".