I believe that the atoning work of Christ properly begins with the Incarnation.
Atonement indicates sacrifice. The worshipper “draws near” the purposed sacrifice, lays his hands on the head of the animal, kills the animal (some miss this point, the animal was killed by the worshipper except when representing Israel…or if the atoning offering was a bird), the priest deals with the blood depending on the sacrifice, and the animal was burned on the altar (some part of the animal was always burnt on the alter as “an aroma pleasing to the Lord”). This foreshadows the Atonement of Christ.
God draws near. Jesus did not regard equality with God a thing to be grasped, but emptied Himself, taking the form of a bond-servant, and being made in the likeness of men. Being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross. Philippians 2:5-8
I believe that the Atonement is both penal and substitutionary. Christ died on the cross in our place, as our representative, bearing our iniquities, so that the justice of God is satisfied and the wrath that we rightfully deserve is averted. The Father offered his Son at the Cross (he did not literally leave him, but he forsook him – he gave him over to Judas, to the Jews, to Pilate, and to the Cross and there offered him as an atoning sacrifice, laying upon him our iniquity).
Galatians 3:13 - Christ redeemed us from the curse of the Law, having become a curse for us—for it is written, "CURSED IS EVERYONE WHO HANGS ON A TREE"—
Isaiah 53:5-6 - But He was pierced through for our transgressions, He was crushed for our iniquities; The chastening for our well-being fell upon Him, And by His scourging we are healed. All of us like sheep have gone astray, Each of us has turned to his own way; But the LORD has caused the iniquity of us all To fall on Him.
Romans 3:23 - for all have sinned and fall short of the glory of God, being justified as a gift by His grace through the redemption which is in Christ Jesus; whom God displayed publicly as a propitiation in His blood through faith.
Hebrews 10:8-10 After saying above, "SACRIFICES AND OFFERINGS AND WHOLE BURNT OFFERINGS AND sacrifices FOR SIN YOU HAVE NOT DESIRED, NOR HAVE YOU TAKEN PLEASURE in them" (which are offered according to the Law), then He said, "BEHOLD, I HAVE COME TO DO YOUR WILL." He takes away the first in order to establish the second. By this will we have been sanctified through the offering of the body of Jesus Christ once for all.
But I also believe it covenantal. Some here either deny the idea or shove it off to the peripheral - slightly out of focus and barely in view.
Genesis 17 speaks of the “Everlasting Covenant” between God, Abraham and his descendants – man's part was circumcision (those without the sign had broken the covenant).
Deuteronomy describes another covenant - Moses took the blood and sprinkled it on the people and said, "Behold the blood of the covenant, which the LORD has made with you in accordance with all these words."
Jeremiah looks to a New Covenant - "Behold, days are coming," declares the LORD, "when I will make a new covenant with the house of Israel and with the house of Judah, not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, ... "I will put My law within them and on their heart I will write it; and I will be their God, and they shall be My people. .. for they will all know Me, from the least of them to the greatest of them," declares the LORD, "for I will forgive their iniquity, and their sin I will remember no more." Jeremiah 31:31-34
New Testament insists that the covenant God made with Abraham is on a different footing than the covenant given at Sinai. Paul links this covenant with the New Covenant as these New Covenant people are the benefactors of the promises made to Abraham (true Israel, true Jews, circumcised hearts). The Atonement ushered in the New Covenant as all covenants involved sacrifice. But the New Covenant is a better covenant, founded on better promises (Hebrews 8:6), and not of the letter but of the Spirit (2 Corinthians 3:6).
Galatians 3:13-14 Christ redeemed us from the curse of the Law, having become a curse for us—for it is written, "CURSED IS EVERYONE WHO HANGS ON A TREE"— in order that in Christ Jesus the blessing of Abraham might come to the Gentiles, so that we would receive the promise of the Spirit through faith.
And when He had taken some bread and given thanks, He broke it and gave it to them, saying, "This is My body which is given for you; do this in remembrance of Me." And in the same way He took the cup after they had eaten, saying, "This cup which is poured out for you is the new covenant in My blood.” Luke 22:19-20
I also believe that the Atonement should be viewed in terms of Jesus’ death accomplishing victory over sin and death. This is a position that some strongly deny, at least strongly deny it should exist except as a footnote to penal substitution (their “primary” view. I do not believe there can be such a thing as a “primary” view. Scripture is scripture). This view of the Atonement takes on the symbolism of the Passover.
I always found it interesting that the first prophesy of the Atonement was on grounds of Christ’s victory.
Genesis 3:15 And I will put enmity Between you and the woman, And between your seed and her seed; He shall bruise you on the head, And you shall bruise him on the heel."
John 12:31-32 "Now judgment is upon this world; now the ruler of this world will be cast out. And I, if I am lifted up from the earth, will draw all men to Myself."
Ephesians 4:7-10 But to each one of us grace was given according to the measure of Christ's gift. Therefore it says, "WHEN HE ASCENDED ON HIGH, HE LED CAPTIVE A HOST OF CAPTIVES, AND HE GAVE GIFTS TO MEN." (Now this expression, "He ascended," what does it mean except that He also had descended into the lower parts of the earth? He who descended is Himself also He who ascended far above all the heavens, so that He might fill all things.)
And there are other passages that I accept as well. I hope that this answers your question and perhaps even highlights our disagreement for future discussion.