The Church of Jesus Christ of Latter-day Saints believe the Aaronic Priesthood was restored not by Aaron but John the Baptist
In answer to their prayer, John the Baptist, under the direction of the Lord Jesus Christ, appeared and conferred upon them the Aaronic Priesthood, saying, “Upon you my fellow servants, in the name of Messiah I confer the Priesthood of Aaron, which holds the keys of the ministering of angels, and of the gospel of repentance, and of baptism by immersion for the remission of sins” (Doctrine and Covenants 13:1). Following John’s instructions, Joseph and Oliver then baptized each other and ordained each other to the Aaronic Priesthood (see Joseph Smith—History 1:68–72.)
Joseph and Oliver then baptized each other and ordained each other: When Joseph Smith and Oliver Cowdery received the Aaronic priesthood, it was conferred upon them slightly different than we practice today. John the Baptist conferred the priesthood upon them, but he did not ordain them to an office. Rather, they were given all the keys of that priesthood, not only the ones pertaining to an office. You may also notice that while the Aaronic priesthood was restored on May 15, 1829, the office of priest as we know it was not fully defined until April of 1830.
So, the authority to baptize each other was given to them when they received all of the rights, powers, and authorities of the Aaronic priesthood. It wasn’t until later that the offices were defined that limit the authority of a person to exercise that priesthood.1
The Aaronic priesthood is prepatory priesthood to Melchizedek.
The Aaronic Priesthood is “an appendage to the greater, or the Melchizedek Priesthood” (Doctrine and Covenants 107:14). As a priesthood holder serves in the Aaronic Priesthood, he prepares to receive the Melchizedek Priesthood, to receive the blessings of the temple, to serve a full-time mission, to be a loving husband and father, and to continue in lifelong service to the Lord.
Hebrews 7 is about superiority of Melchizedek priesthood, not that Aaronic as not needed:
The Melchizedek Priesthood brings exaltation and administers the gospel—It is received with an oath and covenant—The superiority of the Melchizedek Priesthood over the Aaronic Priesthood is explained—Salvation comes through the intercession of Christ.
Hebrews 8:13 is talking about the passing away of the law of Moses not the Aaronic Priesthood.
1 The Restoration of the Aaronic and Melchizedek Priesthoods, Conferring the Priesthood and Ordaining to an Office (See #3 and 4, second list)