Christ was never a member of the Levitical priesthood; he was of the tribe of Judah. He did not function as a priest on earth; he did not serve in the Jerusalem temple, because he is the great high priest of the new covenant who is our Redeemer.
Never forget we need not only an atoning Saviour to suffer in our place, to bear away our sin that will keep us out of hell; we need atonement to save us from punishment, but we also need the perfect righteousness of Christ to deserve heaven for us all. He did both. His blood and righteousness save and secures the two parts of eternal salvation. The better sacrifice, not a slaughtered animal, but the most refined, pure, and wonderful person, the Son of God, who became man. We needed a Saviour who has within him the inherent power of the Prince of life, so that he can raise himself from the dead and ascend into heaven, and resume his station as the everlasting Son of God equal with him, the Father. That is the better sacrifice.