Matthew makes up for Luke's lack?

 One of the most frustrating passages in Luke's Gospel occurs in the story of the Road to Emmaus:

​Luke 24.26-27
Was it not necessary that the Messiah should suffer these things and then enter into his glory?” Then beginning with Moses and all the prophets, [Jesus] interpreted to them the things about himself in all the scriptures.

If only Cleopas, or his companion, had taken notes of Jesus' tour through Scripture.
Something similarly frustrating occurs early on in Mark's Gospel.

​Mark 1.21-22
They went to Capernaum, and when the Sabbath came, he entered the synagogue and taught. They were astounded at his teaching, for he taught them as one having authority and not as the scribes.

When Matthew encounters this omission in Mark he responds with the Sermon on the Mount (inserted into the space between Mark 1.21 (combined with other scene-setting verses) and Mark 1.22).

If Matthew used Luke, as proponents of the MPH propose, then maybe he had a similar reaction in response to Luke 24.26-27? Could this be why Matthew presents virtually every major event in Jesus' life as a fulfilment of Scripture?