Would John the Beloved or the Three Nephites have been able to give Joseph Smith and Oliver Cowdery the Aaronic and Melchizedek Priestnood rather than John the Baptist and Peter James and John?
The Aaronic priesthood is an appendage to the Melchizedek Priesthood and thus can be conferred by anyone in this world holding the Melchizedek Priesthood with one caveat: priesthood ordination is an authorized ordinance. In order for an elder to confer the Aaronic Priesthood on an individual, he must get authorization from that individual’s priesthood leader (usually a bishop or branch president).
It appears that there is a similar order on the other side. When John appeared to Joseph Smith and Oliver Cowdery he said “he acted under the direction of Peter, James and John, who held the keys of the Priesthood of Melchizedek” (Joseph Smith-History 1:72). So presumably these 3 could have performed the ordinance themselves, or commissioned another to do so. “This, then, is the nature of the Priesthood,” taught Joseph Smith, “every man holding the Presidency of his dispensation, and one man holding the Presidency of them all, even Adam; and Adam receiving his Presidency and authority from the Lord” (Teachings of the Prophet Joseph Smith, page 169). And again, “The keys have to be brought from heaven whenever the Gospel is sent. When they are revealed from heaven, it is by Adam’s authority” (page 157).
Even so, it was fitting to assign John the Baptist to give Joseph and Oliver authority to baptize. In the New Testament he is memorialized for baptizing the Savior. The very act is in his name! Peter, James, and John’s selection was very fitting given the principles of dispensation.