Meier says on page 239 of volume 2 of A Marginal Jew: Rethinking the Historical Jesus that the Gospel of Matthew has "Kingdom of Heaven" rather than "Kingdom of God" because Matthew is employing "a pious Jewish periphrasis to avoid constantly naming the Deity in the oblique case of a set formula". When Matthew has "Kingdom of God", which occurs four times in Matthew's Gospel, that is reflecting "the earlier and perhaps original form of the phrase", according to Meier.
I have two questions. First, why are there so many cases in which the Gospel of Matthew uses "God" rather than "Heaven" (see here), if Matthew is using "Heaven" because he is uncomfortable with "God"? Second, why does Matthew follow Jewish periphrasis, according to Meier, when Meier has argued elsewhere that Matthew is not a Jewish Gospel? Can redaction and different layers answer these questions satisfactorily?
Meier states on page 239 that "Matthew's tradition and formation stem from a church that had been strongly Jewish-Christian in its early days." I checked Meier's article on the Gospel of Matthew in the Anchor Bible Dictionary. Meier affirms there that Matthew's use of "Kingdom of Heaven" rather than "Kingdom of God" is a " respectful Semitic circumlocution for God". At the same time, Meier maintains there that the one who put the Gospel of Matthew together was probably a Gentile, for the Matthean redactor is hostile towards Jews, gets things about Judaism and the Hebrew Bible wrong (i.e., he misunderstands parallelism in Zechariah 9:9 and appears ignorant of certain facts about the Sadducees), and drops Mark's Semiticisms.
Meier concludes: "In the end, the theory of Matthew as a gentile Christian who had belonged to the Antiochene church (and perhaps its scribal school) for many years, who revered the Jewish-Christian traditions of his church, and who intended to preserve while interpreting them in his gospel seems to be able to explain all the data more easily." For Meier, the redactor of Matthew was a Gentile who used Jewish-Christian traditions that were at his church.