How is Pitty Sing (the cat) instrumental to the whole plot of "A Good Man Is Hard To Find"?
The cat in O'Connor's "A Good Man is Hard to Find" causes the accident that leaves the family stranded in the vicinity of The Misfit and his partners in crime.
In this O'Connor story, the grandmother must come into contact with The Misfit. The spiritually fallen woman must interact with the twisted, grotesque force that shows her the error of her ways, so to speak. The cat enables O'Connor to steer the grandmother toward the meeting. This leads to the woman understanding, perhaps and at least in part, the truth of her beliefs and behaviors. It also allows The Misfit to reveal the truth that he speaks of when he declares that she would have been a good woman if someone had held a shotgun to her head everyday of her life.
O'Connor believed that this was the only means of salvation for spiritually hardened or lackadaisical Christians.
Incidentally, the cat's presence in the car reveals characterization of the grandmother as well as adding to the plot: it provides still another example of her disrespect for others and her haughty nature.