Harry was sick of the spotlight
Harry's experience as the boy who lived taught him to spurn the spotlight. He was annoyed by the attention of people like the Creevey brothers, and, even worse, the schemers like Skeeter. His fame as the Hogwart's champion in the Triwizard Tournament almost cost him his best friend as well. Harry was very aware that fame can turn quickly to infamy, and he expresses on multiple occasions that he would rather be normal - before the first task in the Triwizard Tournament he dreams of being a spectator without a care in the world, cheering on the Hogwarts Champion.
Mad Eye (Barty Crouch Jr.) set him firmly on the path to become an Auror
Harry really respected Mad Eye, so the latter's suggestion that he would make a good auror stuck with him. Even after the revelation that Mad Eye was an impostor, Harry acknowledges that his complement and career advice lodged in his mind. When the time comes to prepare for O.W.L.s and start exploring career options, he admits he never really considered any other career ever since Mad Eye planted the idea in his head.
It should also be noted that Harry's firebolt was destroyed in book 7, and he likely could not afford another (in book 3 he is concerned that buying one would mean he would have to mooch off of the Weasley's to finish school). While the team he played for would undoubtedly buy him a new broom, this would certainly be a psychological blow to him, especially as it was one of his last connections to Sirius.