Typology grounds not only Christology but soteriology

"… a Christological reading of the Old Testament is not only an effort to discover "snapshots” of Jesus in the history of Israel…David foreshadowed Jesus by his patience in the face of persecution, his sufferings on the way to receiving his kingdom, his kindness to Saul’s house, his victories over Israel’s enemies, his intention to build a house for the Lord, his exile from Jerusalem during the rebellion of Absalom, and in many other more specific ways. On two occasions in 2 Samuel, the death of a son of David preserves David and his kingdom (the first son of Bathsheba; Absalom), and this…points to the cross.

"Illuminating and exciting as these things can be, a Christological reading of David’s life must also operate at another level, It is not merely that David foreshadows Christ in isolated events, but that a Davidic royal theology is developed by the literary-historical presentation of David’s life, and this provides a crucial element of the framework for understanding the ministry of Jesus (that is, it pro­vides a fundamental Christology in the systematic theological sense). …David is presented in Samuel as the new and true Israel. This is true in two senses: First, David’s life followed the pattern of the life of Jacob, the eponym of Israel, and, second, David is the head of the corporate Israel, the representative Israelite, and this is evident in the way his life story both recapitulated the earlier history of Israel and foreshad­owed her future.

But this David-Jacob-Israel connection is understood fully only when it is seen as essential groundwork for the gospels and Paul, where Jesus is seen as the true Israel, recapitulating and redeeming Israel’s history in his own history... the notion that Israel was embodied in the one person of Jesus is crucial to Paul’s Christology and soteriology, and this notion was intro­duced by the book of Samuel. (To be sure, the Aaronic priest was a representative Israelite, but Jesus is more often contrasted with Aaron than compared with him.) Jesus as son of David functions as repre­sentative, and this was made possible by the arrangements of the Davidic covenant. Typology thus grounds not only Christology (who Jesus is—” son of David”) but soteriology (what He accomplished, the ”mechanism” of His atonement, and the effects of His work)." -- Peter Leithart, A Son to Me, p. 16, 17