Admittedly a bit speculative, but here goes:
How does the idea of purification by Jesus' blood tie into theories of atonement?
Heb 1:3 He is the reflection of God’s glory and the exact imprint of God’s very being, and he sustains all things by his powerful word. When he had made purification for sins, he sat down at the right hand of the Majesty on high,
Heb 9:14 how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our conscience from dead works to worship the living God!
Heb 9:22 Indeed, under the law almost everything is purified with blood, and without the shedding of blood there is no forgiveness of sins.
Heb 9:23 Thus it was necessary for the sketches of the heavenly things to be purified with these rites, but the heavenly things themselves need better sacrifices than these.
Reconciliation with God, by establishing the
union with God that man was made for,
means the actual purification of man that the law could never accomplish. The price Jesus paid for bringing the light into the world was His own life. Unlike Adam, He perfectly obeyed God in every way during His life on earth including the death He must suffer to pay for the sins that man committed beginning with and due to that disobedience and the alienation from God that it resulted in.
Man cannot retain moral integrity or wholeness apart from God, not to mention the deep and lasting happiness, peace, satisfaction, etc that we were made to desire and to have. Sin, simply, results in death; it's inconsistent with goodness, with everything that God is. It's certainly never been His intention for us. Separation from God is
already death. Jesus took care of both for us, sin and death. Sin can no longer condemn us to death because we now have the union with God, as Christ
always had, that can overcome sin by its nature, under grace, by the Spirit. Again, that union/communion/direct relationship, entered into via faith, is the primary difference between the old and new covenants.
Our wills are still involved throughout, however, so that we may not
want reconciliation with God-we may not open the door when He knocks, or keep it open to Him later on. It’s a work of His that we participate in as we truly remain in Him-and therefore He in us. He won’t abandon us-but we can abandon Him-and return to death, not interested in purification, not interested in Him.
"Therefore, there is now no condemnation for those who are in Christ Jesus. For in Christ Jesus the law of the Spirit of life set you free from the law of sin and death. For what the law was powerless to do in that it was weakened by the flesh, God did by sending His own Son in the likeness of sinful man, as an offering for sin. He thus condemned sin in the flesh, so that the righteous standard of the law might be fulfilled in us, who do not walk according to the flesh but according to the Spirit." Rom 8:1-4
"Therefore, brothers, we have an obligation, but it is not to the flesh, to live according to it. For if you live according to the flesh, you will die; but if by the Spirit you put to death the deeds of the body, you will live. For all who are led by the Spirit of God are sons of God." Rom 8:12-13
"...not having a righteousness of my own that comes from the law, but that which is through faith in Christ—the righteousness that comes from God on the basis of faith." Phil 3:9