Jesus, being sinless, was the unblemished lamb of the pascha.
There were other animal sacrifices as well— like, for example, the scapegoat, which was ritually burdened with the sins of the populace and then driven away— in addition to the usual Temple sacrifices.
The Aaronic priesthood relied upon animal sacrifices, which were constantly renewed.
Our high priest is Jesus, but he is also our God and our victim, all rolled into one entity. He was able to sacrifice himself, once, for all— and each time we conduct the Mass, we have a re-presentation of that sacrifice.
There’s a lot of discussion amongst historians/theologians about historic animal sacrifice amongst the Hebrew people. There’s one theory that the Hebrews had become accustomed to animal sacrifice during their time amongst the pagans, so when they entered into the Promised Land, and when the Temple was built, all animal sacrifice was centralized to a single location, with God ultimately wanting to move them away from something that had its roots in idolatry. There are a number of verses in the OT where God says through the prophets, over and over and over again, about how it’s not sacrifices he wants, but an obedient spirit. How the smell of sacrifices sickens him until their hearts become pure. That sort of thing.
So yes, I would agree that animal sacrifices are unable to cleanse sin.
If Jesus was able to redeem us merely with his death, the simple thing would have been for him to allow himself to be killed by Herod as an infant. Poof, all done. But instead, he stuck around for 30 years, and then spent 3 years in public ministry. And his most frequent topic was “The Kingdom of God is like…” So that suggests that his preaching was just as important as the mechanics of the Redemption itself.
Out of all the people in the world, Jesus was the one who got to choose not only who he was born to, but also the manner of his death. I suspect that a large part of why he chose to suffer so terribly was to give a visual reminder of how terrible sin itself is, because it separates us from God… and the magnitude of his love in reaching out to redeem humanity.