The Apostles appointed Elders to rule local churches.
It is the apostles that chose and ordained elders in each church. From the elders the apostles chose and ordained bishops.
So there you have the biblical hierarchy in the church. Apostes at the top, that had world-wide authority to grow and govern the church. Then bishops, that had the authority to grow and govern a small area (city) in the world. After the bihop was chosen from the elders, the bishop then could choose and ordain other elders and priests and deacons to assist him in the work in his city. But the bishop could not at the beginning go into another city and claim that city as his to govern spiritually. That was the apostles job, not the bishop.
In all your study to show some kind of succession from apostles being at the head of the church, to bishops being the heads of the churches, you forget the 2 most important scriptures in the bible on this subject:
Ephesians 2:19-20 King James Version (KJV)
19 Now therefore ye are no more strangers and foreigners, but fellowcitizens with the saints, and of the household of God.
20 And are built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone.
Obviously, the word of God is foundational and the apostles spread the word. But more than that, they were part of the foundation of the church hierarchy that anchored the church in the correct word, and the building up of the kingdom of God on earth. (ie chose bishops to grow and govern a specific area of the church)
So the foundation of the church was apostles and prophets and Jesus Christ.
Ephesians 4:11-14 King James Version (KJV)
11 And he gave some, apostles; and some, prophets; and some, evangelists; and some, pastors and teachers.
12 For the perfecting of the saints, for the work of the ministry, for the edifying of the body of Christ.
13 Till we all come in the unity of the faith, and of the knowledge of the Son of God,unto a perfect man, unto the measure of the stature of the fulness of Christ.
14 That we henceforth be no more children, tossed to and fro, and carried about with every wind of doctrine, by the sleight of men, and cunning craftiness, whereby they lie in wait to deceive.
So here is why Jesus chose apostles to oversee the entire world church, and other to assist them in this endeavor.
According to this scripture, without apostles and prophets and others:
1) we cannot be perfected.
2) we cannot do the full work of the ministry.
3) we cannot edify the body of Christ (which are the people). Edify means to teach and instruct spiritually.
4) we cannot grow to maturity in the gospel, we will remain children.
5) we cannot get a handle on the true gospel, but will be tossed to and fro, and carried about with every wind of doctrine.
6) we cannot be protected from the sleight, and cunning craftiness of men as they lie in wait to decieve.
Now another important piece of information: we are to have apostles and prophets and others for how long?
1) Till we all come to a unity of the faith. (has that happened yet?)
2) Until we all come unto a knowledge of the Son of God. (has that happened yet?)
3) Until we measure up to a perfect man, and are equal to the stature of the fulness of Christ.
(has that happened)
So as you can see, the apostles were very important to Jesus, he was leaving his church in their hands. That is why for a time, new apostles were chosen to replace fallen apostles. Until the world and the bishops of the churches rejected them, then the Lord stopped calling new ones to replace old ones, and we see that without them, the perils of Ephesians 4:11-14 came to pass.
Here is a scripture that gives you just a small hint of what I am talking about when I say the bishops of the churches started to reject apostles:
3 John 9-10 King James Version (KJV)
9 I wrote unto the church: but Diotrephes, who loveth to have the preeminence among them,
receiveth us not.
10 Wherefore, if I come, I will remember his deeds which he doeth, prating against us with malicious words: and not content therewith, neither doth he himself receive the brethren, and forbiddeth them that would, and casteth them out of the church.
Was this just one church or was this one of many churches that were starting to be critical of the apostels. Remember the apostles were in charge of the bishops. And when they came to a city they would audit, persay, the church and see if the bishop worthy of his calling. I believe many were not and were excommincated and replaced, which became an issue for bishops trying to use their calling for personal gain. I believe it became widespread.
So the succession of apostles to bishop was not a good thing for the church, it was a disaster.