They Couldn't Prove Who Their Parents Were

These searched among their ancestral registration, but it could not be located; therefore they were considered unclean and excluded from the priesthood.

Nehemiah 7:64

Inclusion in the priesthood was based upon being able to prove one's ancestry. Without the right documents, proof, or whatever, they would be kicked out and considered unclean. This type of system is included in what Paul called "the flesh."

Unless one is born into the right family, they had no chance of getting into the holy place. This dependence upon ancestry is flawed because one's mother or father doesn't dictate the type of person they will be. There were priests, like the sons of Eli, who were pretty awful people.

Now the sons of Eli were worthless men; they did not know the LORD and the custom of the priests with the people. When any man was offering a sacrifice, the priest’s servant would come while the meat was boiling, with a three-pronged fork in his hand. Then he would thrust it into the pan, or kettle, or caldron, or pot; all that the fork brought up the priest would take for himself. Thus they did in Shiloh to all the Israelites who came there.

1 Samuel 2:12–14

Now Eli was very old; and he heard all that his sons were doing to all Israel, and how they lay with the women who served at the doorway of the tent of meeting.

1 Samuel 2:22

Why could these men continue to serve? Because they won the lottery. They happened to be born into the right family. There were hundreds, if not thousands, who could have done the job of Eli's sons without the corruption, but if they went into the Holy Place they would die because they weren't of the priestly lineage.

In the New Covenant, we have no such divisions. Everyone is included in the priesthood regardless of their ancestry, tribe, or gender.

For it is evident that our Lord was descended from Judah, a tribe with reference to which Moses spoke nothing concerning priests. And this is clearer still, if another priest arises according to the likeness of Melchizedek, who has become such not on the basis of a law of physical requirement, but according to the power of an indestructible life.

Hebrews 7:14–16

And so, we are able to draw near to God on the basis of His life and His righteousness. Our ancestry, gender, or social class doesn't matter, only our faith.

Therefore, brethren, since we have confidence to enter the holy place by the blood of Jesus, by a new and living way which He inaugurated for us through the veil, that is, His flesh, and since we have a great priest over the house of God, let us draw near with a sincere heart in full assurance of faith, having our hearts sprinkled clean from an evil conscience and our bodies washed with pure water.

Hebrews 10:19–22