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D&C 107:16 is part doctrine, part prophesy.
The administration of the Aaronic Priesthood doesn't operate by lineage now, but it will one day.
Although the Aaronic Priesthood is conferred in the Church today without restriction to the lineage of Aaron, the keys of this priesthood rightly belong to the firstborn of the seed of Aaron, and in the restoration of all things the office of bishop (president of the priests) will once again be conferred on one of that lineage, as it is designated by revelation to the president of the Church (D&C 84:14β21; 107:13β17).
Bible Dictionary, Aaronic Priesthood
This is part of the more general "Restoration of the Gospel", following the apostasy of the Original Church.
And he shall send Jesus Christ, which before was preached unto you:
Whom the heaven must receive until the times of restitution of all things, which God hath spoken by the mouth of all his holy prophets since the world began.
Acts 3:20-21 (emphasis added)
While this restitution/restoration is often ascribed to events in the 1830s -- like the formation of the LDS Church -- it actually began long before then, and is not yet over.
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My experience is the church does not acknowledge the literal sons of Aaron at all. My husband is a literal son of Aaron and keeps his covenants with God; yet, this has not granted him his legal right (D&C107:76) so far.
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There are a lot of rumors and unsupported assumption surrounding this topic. Here's what we actually know.
The office to which a literal decendant of Aaron is entitled is that of Presiding Bishop. It has nothing to do with the office of Bishop over any given Ward. The calling (of Presiding Bishop) comes through the First Presidency, not through the office of Patriarch normally associated with declaring lineage. The only thing that the rumors got right was that the individual need not call counsellors. "Church History and Modern Revelation, 1:259."
As for holding the Aaronic priesthood, a decendant of Aaron would still have to be ordained. He isn't born with the priesthood. By birth he has the right to the priesthood, not the priesthood itself. To date we have not had any such decendant of Aaron identified and set apart but we believe it will happen.