The usual interpretation of the “mad dog incident” is that the mindless mad dog stands for white racism.
Which if you carry it to its logical conclusion means that you can’t reason with white racists so should just shoot them.
Personally I always thought the incident was put in to illustrate the character of Atticus. He was willing to put himself at risk and take direct physical action when necessary to protect the community. He wasn’t just a bookworm.
Similarly the mean, racist old lady was put in to show that the racist whites were not all evil, that their failings were often counter-balanced by admirable traits. After her death, Atticus called her the bravest person he’d ever known.
In the book, people are complex, with the only totally evil person being Tom Ewell. The local white farmers who made up the lynch mob are not excused, but their basic integrity and honor is shown in various scenes, including the way in which the mob is broken up by Scout.
Most of this complexity got left out of the movie, which made the story much more simplistic.
On a less deep level, I think it showed Jem and Scout that Atticus may not play football for the Methodists, but there was a lot more to him that meets the eye! LOVE the look of astonishment on the face of the kids when he shoots the dog in the movie.
Atticus could have called for the dog catcher. He could have thrown a blanket over the dog. He could have gone inside. He shot the dog. And why a dog and not say, a wild hog? Or a snake? Atticus could kill, and that point is rarely made with students. Why?
TO KILL A MOCKINGBIRD is a huge story for the liberals. It is, of course, fiction......a very useful fiction. What if the dog represented black violence and not white racism?