You are confusing salvation with reconciliation. The activities of the high priest and all the laws of atonement allowed unclean people to be reconciled because they were blood covenant laws.
I am speaking of the corporate provision of salvation that only Jesus could effect through His once-for-all sacrifice of Himself. That is what reconciled people to God:
2 Corinthians 5:14 For the love of Christ controls us, having concluded this, that one died for all, therefore all died;
15 and He died for all, so that they who live might no longer live for themselves, but for Him who died and rose again on their behalf.
16 Therefore from now on we recognize no one according to the flesh; even though we have known Christ according to the flesh, yet now we know Him in this way no longer.
17 Therefore if anyone is in Christ, he is a new creature; the old things passed away; behold, new things have come.
18 Now all these things are from God, who reconciled us to Himself through Christ and gave us the ministry of reconciliation,
19 namely, that God was in Christ reconciling the world to Himself, not counting their trespasses against them, and He has committed to us the word of reconciliation.
20 Therefore, we are ambassadors for Christ, as though God were making an appeal through us; we beg you on behalf of Christ, be reconciled to God.
21 He made Him who knew no sin to be sin on our behalf, so that we might become the righteousness of God in Him.
The activities of the earthly high priest and the blood of animals were ineffectual. They were only shadows that pointed to the reality of Christ, our true High Priest. Only Jesus' blood could really make atonement:
Hebrews 10:1 For the Law, since it has only a shadow of the good things to come and not the very form of things, can never, by the same sacrifices which they offer continually year by year, make perfect those who draw near.
2 Otherwise, would they not have ceased to be offered, because the worshipers, having once been cleansed, would no longer have had consciousness of sins?
3 But in those sacrifices there is a reminder of sins year by year.
4 For it is impossible for the blood of bulls and goats to take away sins.
The law could never reconcile anyone to God; on the contrary, it showed people how far away from God they were and how sinful they were. Romans 5 says that the law "came in so that transgression would increase":
Romans 5:8 But God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us.
9 Much more then, having now been justified by His blood, we shall be saved from the wrath of God through Him.
10 For if while we were enemies we were reconciled to God through the death of His Son, much more, having been reconciled, we shall be saved by His life.
11 And not only this, but we also exult in God through our Lord Jesus Christ, through whom we have now received the reconciliation.
12 Therefore, just as through one man sin entered into the world, and death through sin, and so death spread to all men, because all sinned--
13 for until the Law sin was in the world, but sin is not imputed when there is no law.
14 Nevertheless death reigned from Adam until Moses, even over those who had not sinned in the likeness of the offense of Adam, who is a type of Him who was to come.
15 But the free gift is not like the transgression. For if by the transgression of the one the many died, much more did the grace of God and the gift by the grace of the one Man, Jesus Christ, abound to the many.
16 The gift is not like that which came through the one who sinned; for on the one hand the judgment arose from one transgression resulting in condemnation, but on the other hand the free gift arose from many transgressions resulting in justification.
17 For if by the transgression of the one, death reigned through the one, much more those who receive the abundance of grace and of the gift of righteousness will reign in life through the One, Jesus Christ.
18 So then as through one transgression there resulted condemnation to all men, even so through one act of righteousness there resulted justification of life to all men.
19 For as through the one man's disobedience the many were made sinners, even so through the obedience of the One the many will be made righteous.
20 The Law came in so that the transgression would increase; but where sin increased, grace abounded all the more,
21 so that, as sin reigned in death, even so grace would reign through righteousness to eternal life through Jesus Christ our Lord.
Reconciliation was accomplished only through the death of Christ, not by obeying the law, which no one could do.