The law reserved priesthood for Aaronites, and even God’s Son was not treated as an exception. Jesus belonged neither to Aaron’s family nor to Levi’s tribe. Jesus was from David’s family and Judah’s tribe. The law was so firm that even Jesus could not serve as a priest under the law.
For the One [Christ] of whom these things are spoken belonged to another tribe, from which no one has ever served at the altar. For it is evident that our Lord was descended from Judah, and in connection with that tribe Moses said nothing about priests. (Hebrews 7:13-14)
On earth, Jesus would not be a priest at all.
Now if He [Christ] were on earth, He would not be a priest at all, since there are priests who offer the gifts according to the law. (Hebrews 8:4)
Yet the same chapters of Hebrews show Jesus as our new High Priest. How could this come about? Through three actions taken by God Himself: 1) through the clear call of God as recorded in Psalm 110, and 2) through canceling the law that excluded as priest any non-Aaronite; and 3) through God raising from death His perfect Son to serve as High Priest forever.
For when there is a change in the priesthood, there is necessarily a change in the law as well. For the One of whom these things are spoken belonged to another tribe, from which no one has ever served at the altar. For it is evident that our Lord was descended from Judah, and in connection with that tribe Moses said nothing about priests. This becomes even more evident when another priest arises in the likeness of Melchizedek, who has become a priest, not on the basis of a legal requirement concerning bodily descent, but by the power of an indestructible life. For it is witnessed of Him, “You are a priest forever, after the order of Melchizedek.” For on the one hand, a former commandment is set aside because of its weakness and uselessness (for the law made nothing perfect); but on the other hand, a better hope is introduced, through which we draw near to God. (Hebrews 7:12-19)