Harry’s defining traits, as they have been throughout the series, are bravery, determination, and self-sacrifice. A true Gryffindor, Harry responds to every crisis with courage and resolve. It would simply never occur to Harry to abandon his quest or to choose some other life. Not that he has any viable alternatives. He has no home and no family to go to, he’s wanted by the Ministry, and he can’t go back to Hogwarts. But more important than these considerations, his destiny—to be the boy who defeats Voldemort—is so ingrained in his identity that he can’t imagine trying to avoid it.
However, Harry is not the most focused or relentless hero, at least not until later in the book. His tendency to stray from his quest is not literal or physical, but mental and emotional. When there are no clear leads and nothing to do, Harry cannot command the sort of focus that Hermione can, digging around in books for clues, racking her brains until something occurs to her. At these moments, Harry tends to lose focus and drift, following his emotions. This happens most dangerously in Godric’s Hollow, when Harry leads them into a trap, his real reasons for going there having nothing to do with the quest and everything to do with his grief and doubt concerning Dumbledore.
This doubt is what leads him astray in his quest. Harry is first concerned that Dumbledore was not forthcoming about his own life, and now cannot ask him about it. Then he thinks that Dumbledore didn’t tell him enough to help with the quest, and questions his motives. Finally, he comes to believe that Dumbledore didn’t love him, and that Dumbledore didn’t deserve his love. Harry’s journey is an emotional one, in which he learns to come to terms with the dead, and learns to believe in Dumbledore again so he can complete his quest without his doubts getting in the way.