Why was Jesus not eligible to be a priest after the order of Aaron?
Jesus was not eligible for the Aaronic priesthood because He sprang from the tribe of Judah, not Levi. Since the Levitical priesthood and the order of Aaron were connected to Levi, His Judahite lineage disqualified Him. Instead, He is a priest after the order of Melchizedek, made by the power of an endless life.
Les explains from Hebrews 7 that if perfection were by the Levitical priesthood, under which Israel received the law, there would be no need for another priest to rise after the order of Melchizedek and not be called after Aaron. The answer given is that Gentiles could not approach God through a Jewish high priest; they need a priest who represents Gentiles, as Melchizedek did. The context then says the Lord Jesus sprang out of Judah, so He was not eligible to be a priest after the order of Aaron because He came out of the tribe of Judah. Instead, another priest arises after the similitude of Melchizedek, made not after the law of a carnal commandment but after the power of an endless life, a priest forever after the order of Melchizedek. This connects the Gentile high priest with the Most High God from start to finish.
For it is evident that our Lord sprang out of Juda; of which tribe Moses spake nothing concerning priesthood.