Well, I found this brilliant 10-point summary by Andy Woods explaining why Matthew did write particularly for the Jews in mind.
http://www.spiritandtruth.org/teaching/ ... 72.pdf?x=x
Matthew's Jewish-Christian Audience
Understanding Matthew's overall argument helps elucidate the meaning of Matthew 6:9-13. Although no specific target audience is mentioned, various clues make it apparent that Matthew had a believing Jewish audience in mind.7 The Jewish nature of the book is apparent by noting several factors.
First, the book contains a disproportionate number of Old Testament citations and allusions. Of the book's 129 Old Testament references, 53 are direct citations and 76 are allusions. On thirteen occasions, Christ’s actions are said to be a fulfillment of the Old Testament.
Second, the book follows a fivefold division. The five major sermons of the book are delineated through the repetition of the concluding formula “when He had finished saying these things” (7:28; 11:1; 13:53; 19:1; 26:1). This fivefold structure would have immediately been recognizable to the Jewish mind since Jews had a tendency to categorize items, such as the Book of Psalms and the Pentateuch, according to a fivefold division.
Third, although originally written in Greek, the book evidences a Hebraistic style, parallelism, and elaboration.
Fourth, tote (“then” or “at that time”) reflects a Jewish style. While this term is employed ninety times in Matthew, it is only used six times in Mark, fourteen times in Luke, and ten times in John.
Fifth, the vocabulary of the book is distinctly Jewish. The following Jewish terms are found in the book: David, Jerusalem as the Holy City (4:5; 27:53), city of the great king (5:35), lost sheep of the house of Israel (10:6; 15:24), kingdom of God, and kingdom of heaven.8
Sixth, the subject matter of the topics covered is distinctly Jewish. Among the topics covered are the Law, ceremonial defilements, Sabbath, kingdom, Jerusalem, temple, Messiah, prophecy, prophets, David, Abram, Moses, scribes, Sadducees, and Pharisees.
Seventh, Matthew’s genealogy reveals a Jewish audience. Matthew traces Christ back to David and Abraham rather than back to Adam (Luke 3).
Eighth, Matthew places a special focus upon the Apostle Peter. Because Peter was the apostle to the circumcised (Gal 2:7-8), Matthew’s focus on Peter harmonizes with the Jewish emphasis of his book.
Ninth, unlike the other Gospels that explain Jewish customs to Gentile audiences, Matthew leaves these same Jewish customs unexplained. This is true not only with regard to Jewish rulers (Matt 2:1, 22; 14:1; Luke 2:1-2; 3:1-2) but it is also true with regard to ceremonial cleansing (Matt 15:2; Mark 7:3-4). The customs that Matthew does explain are of Roman rather than Jewish origin (Matt 27:15). Although some of Matthew’s writings seem to anticipate at least some kind of Gentile audience by giving the interpretation of some Jewish words (1:23; 27:33, 46), it does seem to be a general rule that Matthew provides fewer interpretations of Jewish customs than any other Gospel writer.
Tenth, various church fathers, such as Irenaeus, Origen, and Eusebius believed that Matthew wrote to a Jewish audience. Not only was Matthew written to a Jewish audience but to a believing audience as well. In other words, Matthew’s audience primarily consisted of Jewish Christians. Both Eusebius9 and Origen10 indicated that Matthew was written to those within Judaism who came to believe.
