A Priesthood Independent of Ancestry
Heb 7:15-17 And what we have said is even more clear if another priest like Melchizedek appears, one who has become a priest not on the basis of a regulation as to his ancestry but on the basis of the power of an indestructible life. For it is declared: "You are a priest forever, in the order of Melchizedek."
The Levitical Priesthood based on ancestry was shown to be deficient. Such priesthood only lasted a man's lifetime. It was transferred upon death. But the New Covenant, the Melchizedekian order, lasts forever. For Jesus conquered death and lives forever. His priesthood is based upon life.
Heb 7:18-19 The former regulation is set aside because it was weak and useless (for the law made nothing perfect), and a better hope is introduced, by which we draw near to God.
The law, as a set of rules and regulations was useless to change one's behavior. Likewise Paul notes, "Since you died with Christ to the basic principles of this world, why, as though you still belonged to it, do you submit to its rules: "Do not handle! Do not taste! Do not touch!"? These are all destined to perish with use, because they are based on human commands and teachings. Such regulations indeed have an appearance of wisdom, with their self-imposed worship, their false humility and their harsh treatment of the body, but they lack any value in restraining sensual indulgence." Col 2:20-23 This same truth holds for the Law of Moses.
The Law also is useless in that it doesn't justify a person. It only condemns. "No one will be declared righteous in his sight by observing the law; rather, through the law we become conscious of sin." Rom 3:20 "For if a law had been given that could impart life, then righteousness would certainly have come by the law." Gal 3:21b The Law is useless to save people. But "what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering." Rom 8:3
The Law held out no hope and in fact brought us to a place of hopelessness. But under the New Covenant we have a better hope. Christ "abolished in his flesh the law with its commandments and regulations." Eph 2:15 "in order that the righteous requirements of the law might be fully met in us, who do not live according to the sinful nature but according to the Spirit." Rom 8:4