So, when the Old Testament says that animal sacrifices atone for sin, you don’t believe that. You basically don’t believe the Old Testament.
Remember the Passover!
Before there was a Law from Moses to be written down, God had offered a living example so they would understand the principle. The lamb was slain in obedience to the command of God and its blood applied to the lintel and door posts. The result was the Angel of Death sent to exact the punishment of God passed over those protected by the blood of the sacrifice.
However, the blood of THAT lamb did not protect them forever. That sacrifice did not keep death away forever. That entire generation sinned against God and died over the next 40 years in the wilderness. As Hebrews points out, the animal sacrifices needed to be repeated over and over because they only purchased a temporary delay in the execution of God’s justice.
The sacrifices of the OT were to teach men just how terrible our sins were (something had to DIE, because we did not obey) and to illustrate the Perfect Sacrifice promised to redeem mankind from the curse back in Genesis 3. The animals were ALWAYS a place holder for the death of a man ... just a way of buying time until God fixed it.