There is another slant on this, which when I first heard it, made a light come on. It’s all about an ancient blood covenant.
You will recall that when God made the blood covenant with Abraham, that the people would inherit the land, God instructed the patriarch to slaughter animals and then divide them so that there was a path between the carcasses. That night a smoking pot and a torch was seen to pass along the path. (Genesis 15). As I understand the rules around covenant at that time, each person had to pass between the slain animals. Thereby each person was pledging that they would keep their part of the covenant. And the only way to break the covenant was the death of one of the parties. However in this passage Abraham does not pass between the animals, God does – twice. Meaning that God was making a covenant with both Abraham and himself. So if the covenant was broken then God has to honour his pledge, not Abraham. God eventually did this, when he had Moses bring the people out of Egypt.
Later on God told Abraham the sign of the covenant was circumcision – which again is a blood covenant.
After Moses read the law to the people in the desert and when they accepted the law, Moses then sprinkled blood from a sacrifice on them. (Exodus 24) – another covenant sealed with blood.
At the last supper Jesus said of the wine: this is my blood which will be poured out for many, so he was again initiating a blood covenant (albeit with a substitute substance) for the disciples and through them, us.
The physical part of the blood covenant was his death on the cross. So it’s not so much that he dies for our sins, but rather that he was making a blood covenant with God for us. As he was both human and divine he, like the old covenant with Abraham, was making a pledge, both as God and as man.
His physical blood sealed the new covenant and thereby opened for us, our part of the covenant, which is salvation, forgiveness of sins and eternal life. Hence the idea, that he died to save us.