There is a clear Biblical record of it. You just choose to interpret it differently. The Bible is not an exhaustive history of the Church.
Your Elisha analogy fails because we are not aware of any Hebrews claiming this authority through the ages. That is not true of the church. Bishops have been around and recognized since the beginning.
There is no "clear biblical record" of the apostles making new apostles..Jesus never gave them that right or that authority.. He told them to make disciples not apostles.
Jesus chose 12 Apostles for a reason, Twelve had significance to the Jewish people
There were 12 tribes of Israel. We see that carry through in the prophetic book of Revelation
Revelation 21:12 "And had a wall great and high, and had twelve gates, and at the gates twelve angels, and names written thereon, which are the names of the Twelve Tribes of the Children of Israel:"
Revelation 21:14 "And the wall of the City had twelve foundations, and in them the names of the Twelve Apostles of the lamb."
This was not intended to be an ongoing process, the selection belonged to Christ, He never gave that away