I've recently read an interesting book by Michael Gorman, "The Death of the Messiah and the Birth of the New Covenant: A (Not So) New Model of the Atonement." He argues what I've said many times, that Jesus' primary understanding of his death is as bringing the new covenant of Jer. 31:31.
He thinks that this is true of Paul and other NT writers as well, and that this concept can subsume and include others that try to supply explanations for how it works.
"When we read these texts, we find that the new (or renewed) covenant is a transformative, even a creative, act that generates a renewed covenant people of God. More specifically, looking at them as a cluster of thematically interconnected texts, we can say that this new-covenant people will have several characteristics. The community of the new covenant is:
* liberated (having experienced a new Exodus);
* restored and unified (Israel and Judah together;
* gathered from the peoples; returned to the land of Israel;
* one heart);
* forgiven, cleansed from unholiness and idolatry/infidelity to YHWH;
* sanctified; existing in a mutual covenantal relationship with YHWH (“I will be their/your God, and they/you shall be my people”) characterized by community-wide faithfulness, intimacy, and knowledge;
* internally empowered and enlivened (law / new spirit / God’s Spirit within; heart of flesh, not stone) to keep the Law/covenant;
* bearing witness to YHWH’s holiness;
* experiencing shalom: at peace with God and secured from enemies;
* and permanent, i.e., partners in an everlasting covenant.
That's about as much as I can reasonably give in a posting.