I think there might be some confusion with the word “succession” and “apostle” on these posts. We do see offices in the church and the qualifications speciifcally for them. Offices such as bishop and deacons. We don’t see any such office for an apostle though. An apostle was a one time and unique postition in the NT church that had requirements that are no longer possible to fulfill for example. We can see this in Acts 1:21-22. These are conditions that cannot be fulfilled today. If you want to use the term for bishop or deacon succession i don’t think there is much of a problem with that since the qualifications can be fulfilled in certain men. For an apostle though they cannot be.
No, I think we are all on the same page with this. When we speak of the Apostolic Succession, we are speaking of those persons appointed by the Apostles, invested with the same authority that Christ invested in the Apostles, and that authority being passed on through the laying on of hands to each succeeding generation. We see this was already the practice in the early church:
Early Church Fathers on Bishops, Priests and Deacons
“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 further provision that, if they should die, other approved men should succeed to their ministry.” (Pope St. Clement of Rome, Epistle to the Corinthians, 42:4-5, 44:1-3 [80 A.D.])
“Be subject to the bishop and to one another, as Jesus Christ was subject to the Father, and the Apostles were subject to Christ and to the Father; so that there may be unity in both body and spirit” (St. Ignatius of Antioch, Letter to the Magnesians, A.D.110, [13,1])
“Take care to do all things in harmony with God, with the bishops presiding in the place of God and with the presbyters in the place of the council of the Apostles.” (St. Ignatius of Antioch, Letter to the Magnesians, A.D. 110, [6,1])
“Those, indeed, who belong to God and to Jesus Christ - they are with the bishop. Do not err, my brethren: if anyone follow a schismatic; he will not inherit the kingdom of God.” (St. Ignatius of Antioch, Letter to the Philadelphians, A.D. 110, [3,2])
“You must all follow the bishop as Jesus Christ follows the Father, and the presbytery as you would the Apostles. Let no one do anything of concern to the Church without the bishop. Wherever the bishop appears, let the people be there; just as wherever Jesus Christ is, there is the Catholic Church.” (St. Ignatius of Antioch, Letter to the Smyrnaeans, A.D. 110, [8,1])
“Indeed, when you submit to the bishop as you would to Jesus Christ, it is clear to me that you are living in the manner not in the manner of men but as Jesus Christ, who died for us, that through faith in His death you might escape dying. It is necessary, therefore, - and such is your practice, that you do nothing without the bishop, and that you be subject also to the presbytery, as to the Apostles of Jesus Christ our hope.” (St. Ignatius of Antioch, Letter to the Traillians, A.D. 110, [2,1])
“And of the elect, he was one indeed, the wonderful martyr Polycarp, who in our days was an apostolic and prophetic teacher, bishop of the Catholic Church in Smyrna. Every word which came forth from his mouth was fulfilled and will be fulfilled.” (Martyrdom of Polycarp 16:2, [155 A.D.])
“Although he [Paul] writes to the Corinthians and to the Thessalonians for their correction, nevertheless it is shown that there is one Church spread abroad though the whole world.” (Muratorian Fragment, [155 A.D.])
“Let us be careful, then, if we should be submissive to God, not to oppose the bishop.” (St. Irenaeus, Against Heresies, A.D. 180, [5,3])
“And we are in a position to enumerate those who were instituted bishops by the Apostles, and their successors to our own times: men who neither knew nor taught anything like these heretics rave about.” (St. Irenaeus, Against Heresies, A.D. 180, [3,3,1])