The solution Paul develops begins with the martyr's death which atones for the sins of Israel. But with Paul, we discover this is universalized. The Gentiles have been grafted onto the tree. The atonement is efficacious. It is not something earned by righteous behavior. It is a benevolent act accomplished by God through a divine being Jesus, the one who was equal with God but emptied himself to be born in the likeness of a man (Phil 2:5-7). This benevolent act was part of a plan that God had put into place to defeat Satan and his army of angels that had been battling against the archangels since the beginning of time. The defeat of these cosmic powers and authorities began when they crucified Jesus (1 Cor 2:6-7; Col 2:15; Eph 6:12; cf. 1 Cor 15:23-28). This is what Ephesians is about (although we can argue if Paul wrote it or someone else).
This cosmic understanding of Jesus' death as the beginning of the defeat of the demonic powers that rule the world meant that there had to be a ritual shift too. So baptism wasn't just about cleansing from sins. It became about dying with Jesus. Paul reasoned that if at baptism you were cleansed and received the holy spirit by invoking Jesus' Name, the spirit one received must be Jesus' own spirit. Because the person was possessed by the Christ, the person became Jesus participating in his death and its atonement and provisionally resurrected as Jesus was.
This shift is also evidenced in the performance of the eucharist. It could no longer be a Messianic party with joy and celebration in anticipation of paradise. It became about ingesting a sacrificial meal, about reacting the death of Jesus as a sacrifice for all gathered.