This may sound like a controversial opinion at first, but I need you to hear me out a little bit.
The seven years that Harry was immersed within the World of Sorcery* were tumultuous and traumatizing. In his very first year he killed a professor and thwarted Tom Riddle for the second time in his life. In his second year, he did so again. He spent almost the entire next school year under the impression that the man that betrayed his parents would be back to finish him off as well. When he was finally given hope for leaving his abusive life at the Dursleys, it is taken away just a half-hour later. Then, if that wasn't enough, his best friend turns on him for something he did not do. This is the first of many occasions in which Harry tells the truth, but the people he is close to do not believe him regardless. He's then thrust into a series of trials which no man has completed for centuries, risking his life, and the moment his hands touch glory for his accomplishment, he witnesses the death of the friend that helped him achieve it. Furthermore, he witnessed Voldemort's rise to power, something that would ensure that he never lives a peaceful life at Hogwarts again. The next year is the book that is the most difficult to read, and yet the most enthralling. You can feel Harry's anger and confusion and frustration throughout. Dolores Umbridge literally tortures him in detention (something that he undoubtedly should have wrote to the Ministry about, as this sort of punishment is definitely unsanctioned, Snape tortures Harry as a method of teaching (necessary, but I digress), and his connection to Voldemort causes him to have painful nightmares. To add insult to injury, Sirius Black perishes at the hands of Bellatrix LeStrange. She takes away the one thing that Harry had to look forward to in his life: life away from the Dursleys. While Sirius Black's exonneration is warranted and probably should have helped, it almost adds insult to injury for the Ministry to admit to being wrong too late. Then, the next year, Harry's mentor and father-figure, Albus Dumbledore have a connection over the year that climaxes in his murder at the hands of someone he presumed to finally be innocent.
Then his next year was entirely devoted to killing the man that ruined his life, that causes all of this pain and trauma and suffering towards him and wins. It's finally over. Harry doesn't have to worry about Tom Riddle any longer.
Argument 1: Harry wants to make sure that all dark wizards from here on out get justice for their actions. He said he wanted to be an Auror.
Argument 2: Harry is tired and just wants a relaxed life away from dark wizards and everything wrong with the world. Finally he has peace. Why would he give it away for much smaller threats in the world? No living person will ever stoop to the level of Voldemort, and thus the excitement will be gone for good. All Harry has ever wanted was a normal life.
I think that it makes much more sense for Harry to be offered the position as Professor of Defense Against the Dark Arts either immediately, or after another term at Hogwarts with a substitute. However, I think it would be rather poetic if Harry was immediately instated as Professor because it's the position that Riddle wanted and could never get, and that no one from then on was able to keep. Harry killed Voldemort, and thus the jinx will be broken by him as well.
I had always assumed that this would be the path that Harry would take anyway, considering that he taught his fellow students and members of Dumbledore's Army--and well, might I add.
Argument 3: Well then why didn't he tell McGonagall that that's what he wanted to do?
Counterargument: Because he hadn't yet the experience of trying to find and kill a Dark Wizard, and he wasn't all that accepting of the fact that he was a good teacher anyway. I think that Harry seeing all of his students in the DA fighting for Hogwarts reminded him that he did well.
I think that Harry is Professor of Defense Against the Dark Arts, not an Auror.
*I think the term World of Sorcery or World of Magic is a bit more inclusive than Wizarding World. Wizard is meant to be the male equivelant of the word Witch, right?