I think the first person helps us relate to a place and sequence of events most of us have no experience with. It also helps us to understand what the characters are going through on a more personal level. It makes us feel like we are experiencing it with Amir.
An author makes a critical decision when choosing the narrative perspective from which a story will be told. One type of point of view is no better or worse than another--the decision is based on what the author wants to communicate to the reader. Given this, if the point of view of The Kite Runner is changed, then the message that the novel communicates to the reader will also be changed. The first person narrative perspective of Amir allows the reader to really get into his thoughts and feelings regarding his familial situation and his feelings about betraying Hassan. One of the major themes of the novel is about personal redemption, and the reader walks through this journey as Amir battles with trying to atone for his past mistakes. If the point of view changes, the events of the story might very well be the same, but the reader would lose the inner connection with Amir.
This is an interesting question because so much depends on the reader's own opinion on whether Amir's narration works. For the most part, Amir is a reliable narrator, even if he's a flawed human being. Readers witness events as they happen--Amir recounts the story and comments on his experiences. Because Amir is reliable the reader accepts, without question, the statements of fact and judgement. In fact, the use of first person humanizes Amir. We understand he's flawed as a boy and we follow his maturation process through his return to Afghanistan.
If Hosseini would have chosen, say, an unreliable narrator, the reader would have difficulty believing Amir's recollection of events and experiences; we would be forced to qualify Amir's statements and his collective memory.
Since Hosseini wants to 'educate' Westerners about his native culture and the life of the Afghan people, the story must be told in the first person and from a reliable narrator. It's the only way readers can build empathy for Amir and for the beautiful culture that was/is Afghanistan.