Now let us examine the priesthood of Israel. The key section of Scripture on this subject is found in Hebrews 7:11-28. In this portion of Scripture the author of Hebrews is making the case that the priesthood of the Old Covenant did not do anything. By that I mean that the sacrifices prescribed by the Mosaic Law could not take away sins. They only made the Israelites ceremonially clean. This means that the sacrifices of the Old Covenant era only brought the Israelite into good standing with the nation of Israel and not with the God of Israel.
If perfection could have been attained through the Levitical priesthood (for on the basis of it the law was given to the people), why was there still need for another priest to come—one in the order of Melchizedek, not in the order of Aaron? (Hebrews 7:11 NIV)
The blood of goats and bulls and the ashes of a heifer sprinkled on those who are ceremonially unclean sanctify them so that they are outwardly clean. (Hebrews 9:13 NIV)
But those sacrifices are an annual reminder of sins, because it is impossible for the blood of bulls and goats to take away sins. (Hebrews 10:3-4 NIV)
The priesthood of the Old Covenant functioned only as a picture and not the real thing. As a priesthood it prefigured the coming priesthood of Jesus which is called the Melchizedek priesthood.
For it is declared: “You are a priest forever in the order of Melchizedek.”(Hebrews 7:17 NIV)
Of the priesthood of Jesus the author of Hebrews says that this priesthood actually accomplishes the saving of a people by the death of Jesus on the cross, which is the work of a priest who offers up a sacrifice, in this case his own death, for all those for whom he represented, in this case all those whom the Father chose to save.
Now there have been many of those priests, since death prevented them from continuing in office; but because Jesus lives forever, he has a permanent priesthood. Therefore he is able to save completely those who come to God through him, because he always lives to intercede for them. (Hebrews 7:23-25 NIV)
With the coming of Jesus the priesthood of Jesus took that place of the priesthood of the Old Covenant. His priesthood purchased a people who are also described as being a part of his priesthood.
But you are a chosen people, a royal priesthood, a holy nation, a people belonging to God, that you may declare the praises of him who called you out of darkness into his wonderful light. Once you were not the people of God; once you had not received mercy, but now you have received mercy. (1 Peter 2:9-10 NIV)
In the New Covenant the work of Jesus as our high priest secured our salvation by his death on the cross and purchased for us an eternal priesthood. This means that we now become the real people of God, eternally forgiven, unlike Israel who are only a picture of the people of God. The priesthood of the Old Covenant could not produce a real people of God because it could not take away sins. It is only the priesthood of the New Covenant that truly pays the penalty for the sin for all for whom it was intended. The church does replace Israel for it is only the church that has experiences the satisfying of the wrath of God. The church is the true Israel of God.