No inner completion ever came from the part of a man bound to law, succession, and ritual. That is why a higher order rises without any lineage at all. Melchisedec is the Christ consciousness, the right hemisphere of the brain, present before Abraham and needing no priesthood to install it. It stands already inside a person, older than any ceremony the lower mind can build.
Levi is the faculty joined to outward moral rule, obedient and bound to what is passed down; Aaron is the light, another part of the mind set to minister. But wholeness does not descend through either of them. It is inborn in the higher self, the right side of the brain that was always the true priest, and it wants no succession to make it real.
Melchisedec: the Christ consciousness, the right hemisphere of the brain, present before Abraham
Aaron: the light, another part of the mind given to ministering
Levi: the part of a person joined by outward attachment to moral law and succession