The concept of a Messiah and his relationship- was uppermost in the minds of Jesus’ hearers.
NT Wright states:
“One well-worn but unsustainable position is that the “Jewish background is a mass of legalism and formalism, and that Jesus can teach a different religion, namely an interior spiritual sort. This is clearly no good. If it were true, Jesus would have been simply incomprehensible, a teacher of abstract and interior truths to a people hungry for God to act within history. The people were asking for bread and freedom, not thin air.”
And central to God acting in history was the concept of a Messiah and Jesus self-understanding. The genealogy firmly roots us in history, the here and now, not as some sage/philosopher but as king. King from the royal tree of David, legal heir, and entitled to the prerogatives of rule.
Kingdom of Heaven (KOH) mentioned over 30x, central theme. God the Father working through His Messiah to establish His will on earth as it is in heaven. Mat 6:10
v16 and Jacob the father of Joseph the husband of Mary, of whom Jesus was born, who is called Christ.
Jesus is not born of Joseph by natural means but legal ones and is thus a true Davidic heir.
ἰησοῦ χριστοῦ, Jesus Christ Jesus is personal name
Christ, anointed, Hebrew Messiah
Messiah king stirs the hopes that is God acting in history to save His people
The son of David, the son of Abraham note the inversion of chronological ordering. Harkening to Augustine’s principle of not altering the revelation given, the listing of David first evokes the great promises given to David and his descendants.
2 Sam 7:12-17 firm kingdom lasting forever ruling over other nations, with all nations being blessed through him (harkening to Abraham’s covenantal promises).
Son of David was popular phrase in ANE as a synonym of Messiah.
Son of Abraham roots Jesus squarely with the chosen people with promises of blessing and multiply with worldwide blessings Gen 12:2, Gen 17:2-9
The Magi prostrating before the Child & His Mother, offering gifts, coming from the east ties into this concept of world-wide blessing.
Another highlight of Matthew’s linage is the inclusion of non-Israelite women in the royal line. (Tamar, Rahab, Ruth, wife of Uriah) This underscores God’s concern for the Gentiles and the universal mission of Jesus. We note the difference in conception between Bathsheba and Mary.
Composing an artificial hagiography both the good and bad are listed. Noting is whitewashed. God works through the weakness of men. Peter’s flaws are not hid, part and parcel of the story.
 N.T. Wright, Jesus and the Victory of God (Fortress Press: Minneapolis 1996) 92
 Wright 94