.."looking" at scripture to see how the NT writers presented this topic -- was it of high concern or did it not make even honorable mention,, or was it not actually a thing at all?
It is not an argument to say "no successor can be chosen"
The question from scripture is to note how it compares. IS this a big topic where they are introducing apostolic succession in a new religion and getting everyone on board with this new idea? Or does it not even make honorable mention even if it was being done.
If it was not worthy of mention - but was being done... why is that?
The NT church was expecting the imminent return of Christ. So setting up a church in perpetuity was not an issue they specifically addressed. As Albion referred to, it became an issue in the late first century and early second century (post New Testament).
otoh, There is a New Testament model given for how new leaders and successors were chosen and "ordained."
For example:
deacons
"Therefore, brethren, select from among you seven men of good reputation, full of the Spirit and of wisdom, whom we may put in charge of this task. "But we will devote ourselves to prayer and to the ministry of the word." The statement found approval with the whole congregation; and they chose Stephen, a man full of faith and of the Holy Spirit, and Philip, Prochorus, Nicanor, Timon, Parmenas and Nicolas, a proselyte from Antioch. And these they brought before the apostles; and after praying, they laid their hands on them. Acts 6:3-6
Bishops
Paul and Barnabas appointed elders for them in each church and, with prayer and fasting, committed them to the Lord, in whom they had put their trust. Acts 14:23
For Paul
Then Ananias went to the house and entered it. Placing his hands on Saul, he said, "Brother Saul, the Lord—Jesus, who appeared to you on the road as you were coming here—has sent me so that you may see again and be filled with the Holy Spirit." Immediately, something like scales fell from Saul's eyes, and he could see again. He got up and was baptized, and after taking some food, he regained his strength. Acts 9:17-19
Paul to Timothy
For this reason I remind you to kindle afresh the gift of God which is in you through the laying on of my hands. 2 Tim. 1:6
Do not neglect the spiritual gift within you, which was bestowed on you through prophetic utterance with the laying on of hands by the presbytery. 1 Tim. 4:14
There could be more examples.
In each case, the existing Apostles/presbytry appoint the new/subsequent clergy with prayer, fasting and the laying on of hands.
Then there is the early Church witness:
"Through countryside and city [the apostles] preached, and they appointed their earliest converts, testing them by the Spirit, to be the bishops and deacons of future believers. Nor was this a novelty, for bishops and deacons had been written about a long time earlier. . . . Our apostles knew through our Lord Jesus Christ that there would be strife for the office of bishop. For this reason, therefore, having received perfect foreknowledge, they appointed those who have already been mentioned and afterwards added the further provision that, if they should die, other approved men should succeed to their ministry" First Epistle of Clement Letter to the Corinthians 42:4–5, 44:1–3 A.D. 80
"When I had come to Rome, I [visited] Anicetus, whose deacon was Eleutherus. And after Anicetus [died], Soter succeeded, and after him Eleutherus. In each succession and in each city there is a continuance of that which is proclaimed by the law, the prophets, and the Lord" Hegesippus Memoirs, cited in Eusebius, Ecclesiastical History 4:22 A.D. 180
"It is possible, then, for everyone in every church, who may wish to know the truth, to contemplate the tradition of the apostles which has been made known to us throughout the whole world. And we are in a position to enumerate those who were instituted bishops by the apostles and their successors down to our own times, men who neither knew nor taught anything like what these heretics rave about" Irenaeus Against Heresies 3:3:1 A.D. 189
"[The apostles] founded churches in every city, from which all the other churches, one after another, derived the tradition of the faith, and the seeds of doctrine, and are every day deriving them, that they may become churches. Indeed, it is on this account only that they will be able to deem themselves apostolic, as being the offspring of apostolic churches. Every sort of thing must necessarily revert to its original for its classification. Therefore the churches, although they are so many and so great, comprise but the one primitive Church, [founded] by the apostles, from which they all [spring]. In this way, all are primitive, and all are apostolic, while they are all proved to be one in unity" Tertullian Demurrer Against the Heretics 20 A.D. 200
