Q. How come only five disciples of Jesus Christ wrote books in the New Testament? My theory is that for one thing John and Peter were closer to Jesus. Matthew was a Levite from the priestly tribe of Levi, making his role that of writing on Christ’s priesthood. Christ redeemed the priesthood of Levi back unto himself and redeemed Matthew the tax collector from what was considered a disgraceful and corrupt profession. But I don’t know about the others.
I think your question actually contains a good start on its own answer. But first, let me say that if we accept the traditional understandings of authorship, only three of Jesus’ disciples wrote books in the New Testament. You mention John, who is traditionally credited with the Gospel of John, the three letters of John, and the book of Revelation. Two letters that Peter wrote are in the New Testament. And then there is Matthew.
But the James who wrote a book in the New Testament is not the James who was a disciple of Jesus. Rather, he was one of Jesus’ brothers. So was Jude, who wrote another book. Luke and Paul, the other remaining authors whose identities we know, were similarly not among the original twelve disciples. We don’t know who wrote the book of Hebrews, but many things about it suggest that this was someone from the Jewish community in Alexandria, Egypt (whether or not the book was actually written there), so its author was likely not one of the disciples either.
But let us return to Matthew, who, as you noted, was most likely a Levite. (In fact, in relating the same episode in which he is called Matthew in the gospel by that name, Mark calls him Levi. This might have been a nickname or surname; either way, it identifies him with that tribe.) I would say that Matthew’s gospel does more than speak of Christ’s priesthood; it explains the significance of his whole life and ministry as the Jewish Messiah, and his sacrificial death, against the background of the Hebrew Scriptures. As a result, Matthew’s gospel has far more quotations from and allusions to the Old Testament than the other three gospels. So it is a book written for a Jewish audience by someone who was deeply versed in the Jewish Scriptures.
All the other disciples were Jewish as well. We can imagine that they might well have addressed a similar audience in a similar way. And that would have limited the reach of the New Testament, which is the story of how Jesus brought the work that God had done to that point, as described in the Old Testament, to its culmination for the benefit of the whole world.
And so other types of authors were needed, to write to other audiences in other ways. Luke was a Gentile, and he wrote in excellent Greek to a Greek audience. His two works, Luke and Acts, make up a quarter of the New Testament.
Paul was Jewish, in fact, he was a trained rabbi, but he came from Asia Minor, from a context outside of Palestine that was Greek in language and culture. So was also familiar with Greek philosophical thought, as we can tell from his own writings, and from his speeches that Luke records in Acts. Paul writes largely to Christian communities made up of both Jews and Gentiles. His letters comprise another quarter of the New Testament.
While Mark was Jewish, he wrote his gospel in Rome, and we can tell that he is addressing a Roman audience. (For one thing, he uses many Latin terms, and he also explains customs for his readers that a Jew living in Palestine would have understood implicitly.) John was also Jewish, but he likely wrote his gospel in western Asia Minor, and while he refers extensively to the Jewish background of Jesus’ life and ministry, he speaks in a way that is accessible to the broad population of the empire. And as I have already noted, the book of Hebrews likely comes from the Alexandrian context.
So most of the New Testament actually comes from outside the Palestinian Jewish context in which Jesus and his disciples operated. But this allows the New Testament books to speak to a much broader and wider audience than they would have if most of them had been written within that context instead. So, as I said, your reflections about Matthew pointed in the direction of what I think is the answer to your question. Certainly a gospel like his was needed to interpret the meaning of Jesus for a Jewish audience. But the New Testament needed to speak to many other audiences as well, and that is why the authors of most of its books appropriately come from a range of contexts and backgrounds, not only the original circle of Jesus’ twelve disciples.
Good Question 
Luke was a Gentile, and he wrote in excellent Greek to a Greek audience. His two works, Luke and Acts, make up a quarter of the New Testament. Paul was Jewish, in fact, he was a trained rabbi, but he came from Asia Minor, from a context outside of Palestine that was Greek in language and culture. So was also familiar with Greek philosophical thought, as we can tell from his writings and from his speeches recorded in Acts
I don’t know if you can edit this section but it says Luke and Paul wrote Acts.
I can see how that would have seemed unclear. I have now edited the post to make it clearer that Luke wrote Acts and Paul wrote other letters. Thank you.
You mentioned that you didn’t know who wrote Hebrews. Paul wrote Hebrews and many of the books in the New Testament.
Actually, while it is traditional to believe that Paul wrote Hebrews, the consensus of biblical scholarship is that he most likely did not. The author of the book is unknown.
Off topic:
Why it was Eve who was tempted by the devil and not Adam?
Both Adam and Eve were tempted by the devil. Adam was present while Eve was speaking with the serpent. The book of Genesis says, “When the woman saw that the fruit of the tree was good for food and pleasing to the eye, and also desirable for gaining wisdom, she took some and ate it. She also gave some to her husband, who was with her, and he ate it.” So maybe the question is actually why the serpent struck up a conversation with Eve rather than with Adam. I think the reason is that Adam heard the command about the fruit directly from God, while Eve heard it secondhand from Adam, and she believed that the command was not to eat the fruit and not even to touch it. That was going beyond what God had said, and the fact that Eve believed God had said this made it easier for the serpent to portray God as stingy and jealous for his own prerogatives.