They had no printing presses in the late first century. You are assuming an early wide distribution of Christian texts that requires evidence.I believe the scholarly consensus makes less sense than Luke reading Matthew and deciding he needed to write a different Gospel to serve his own agenda.
The scholarly consensus identifies the sayings source Q as the major sayings source used by Matthew and Luke, but not by Mark or John. Your theory is not consistent with the existence of Q. The claim that the sayings shared by Matthew and Luke stem from Luke's use of Matthew can be refuted by all the cases where Luke preserves the more original wording: e. g.
(1) "Whoever comes to me and does not hate father and mother...cannot be my disciple *(Luke 14:26)."
"Whoever loves father and mother more than me is not worthy of me (Matthew (Matthew 10:37).
Luke preserves the Hebrewism, the use of "hate" not as a venomous emotion, but as an expression of priorities. Matthew cleans up the saying for his Gentile readers to remove the offense, while capturing the intended meaning.
(2) Matthew expands and spiritualizes Luke's 6 beatitudes ("poor" into 9 (Luke 6:20-23 // Matthew 5:3-12). Matthe spiritualizes "poor" (Luke) into "poor in spirit" and "hunger now" (Luke) into "hunger and thirst for righteousness."
All scholars agree that Luke has truncates the Lord's Prayer (11:1-4) preserved in its original form in Matthew 6:9-13, whose wording is attested in Didache.
(3) Why would Luke's infancy narrative omit the story of the magi following the Star of Bethlehem? And why would Luke omit Matthew's Galilean resurrection appearance and the Great Commision that the Risen Lord authorizes there?
More importantly, you bear the burden of proof to buck the scholarly consensus and identify evidence for Luke's dependence on Matthew.
The sayings in the Gospel of Thomas are structured after the pattern of Valentinian Gnostic initiation formulae from around 140 AD. (Princeton paid me $1,000 for my painstaking demonstration of this pattern in my Master's thesis.) Apart from the dating problem, Luke would hardly use a heretical Gnostic sayings source for his Gospel. He used Q instead.They both might have also referred to the gospel of Thomas.
There is not a shred of evidence for John's awareness of the 3 Synoptic Gospels. Indeed, Cambridge U professor C. H. Dodd wrote a book demonstrating that where John and the Synoptics run parallel, John preserves the more historically accurate version. In fact, most scholars explain the few parallels in terms of overlapping oral tradition that crept into John's sources.John probably had copies of all three synoptic Gospels and saw little need for another biography. The Gospel of John is a set of theological claims served in a carefully designed story. The I AM claims and the miracles are carefully designed components of an argument.