Matthew was written by Matthew to all the Jewish people. He used over 65 references to the Old Testament. Only the Jewish people would have understood the references. This Gospel was about the man, Jesus
Mark was written by Mark to the Romans. The Romans believed in might and power. Mark wrote to the new Christians telling of Jesus might and power.
Luke was written by Luke to the Greeks. The Greeks were into the body. Luke, as the physician, was the perfect one to tell the new Christians about the "body" of Christ.
John was written by John to the Jews and the new Christians. It was written to show the deity of Christ.
Yet, throughout the first 3 Gospels, the story of Christ's birth and ministry are similar in many respects. Each gave their view of what they saw Jesus do.
You pose an interesting point of view, but most Bible scholars (I am not one, but I have read a number of their books) would likely diagree with your thesis.
The first three Gospels in the New Testament are sometimes referred to as the "Synoptic Gospels." This is because because of their similarities, not only in content, but in some stylistic and linguistic details (analysis of unique words and phrases, common words and phrases, etc). Many, if not most, new Testament scholars believe that the Gospel according to Mark is the oldest of the Gospels (Gospel is from the greek word for "good news"). It is thought that Mark was a follower and scribe of Peter and that the book recalls Peter's teachings. Mark is the shortest of the first four Gospels, and except for the bit (Mark 16: 9-20), is probably in tact. (It is widely held that that part was "added" on by another writer - it is not present in th early Greek texts. Catholics are instructed that all of Mark is canon, but that the "longer ending" need not be considered the work of Mark himself.) The Book of mark probably had it origins around 60-80 AD
Almost all of Mark is contained in the Gospels according to Matthew and Luke. Matthew was probably first recorded in Hebrew or Aramaic, and was later translated into Greek and Latin. Early church leaders (c. 150-200 AD) were familiar with the Gospel of Matthew. Matthew may be a reiteration of Mark with additional detail. It may also have borrowed from a contemporaneous Gospel, that of Luke.
Luke was quite likely and early disciple of Paul. Luke is also thought to be the author of the Acts. It is quite likely that Luke was written in Greek. Like the Book of Matthew, The Book of Luke contains many passages from the earlier Mark, and shares some similarities (stylistic and linquistic) with Matthew.
There is additional detail is the Synoptic Gospels which suggest another source, sometime referred to as "Q." Q may be something like the apocryphal Book of Thomas (a list of purported sayings of Christ, may of which are found in the Gospels, but many that are not).
In short, the idea that the four Gospels were written by four authors may be truer than you think, but that each Gospel was the work of a single author is not supported by 1800 years of research.