Your entire long post was wasted because the entire premise is based on untruth. First off, there is no proof that Peter was actually a Bishop in the Roman church. Second, There is no proof that Linus was even a Bishop anywhere. Let's look at a couple of statements about that from historians.
Very little is known about Linus. St. Irenaeus of Lyons (d. 200) and the historian Eusebius of Caesarea (d. ca. 339) identified him with the companion of Paul who sent greetings from Rome to Timothy in Ephesus (2 Timothy 4:21), but Scripture Scholars are generally hesistant to do so...It should be remembered that contrary to pious Catholic belief--that monoarchical episcopal structure of church governance (also known as the monarchical episcopate, in which each diocese was headed by a single bishop) still did not exist in Rome at this time (McBrien, Richard P. Lives of the Popes: The Pontiffs from St. Peter to Benedict XVI. Harper, San Francisco, 2005 updated ed., pp. 33-34).
Richard P. McBrien was a priest and editor of the Catholic Encyclopedia.
The "Liber Pontificalis" asserts that Linus's home was in Tuscany, and that his father's name was Herculanus; but we cannot discover the origin of this assertion. According to the same work on the popes, Linus is supposed to have issued a decree "in conformity with the ordinance of St. Peter", that women should have their heads covered in church. Without doubt this decree is apocryphal, and copied by the author of the "Liber Pontificalis" from the first Epistle of St. Paul to the Corinthians (11:5) and arbitrarily attributed to the first successor of the Apostle in Rome. The statement made in the same source, that Linus suffered martyrdom, cannot be proved and is improbable. For between Nero and Domitian there is no mention of any persecution of the Roman Church; and Irenaeus (1. c., III, iv, 3) from among the early Roman bishops designates only Telesphorus as a glorious martyr. Finally this book asserts that Linus after his death, was buried in the Vatican beside St. Peter. We do not know whether the author had any decisive reason for this assertion...There was nothing in the liturgical tradition of the fourth-century Roman Church to prove this, because it was only at the end of the second century that any special feast of martyrs was instituted and consequently Linus does not appear in the fourth-century lists of the feasts of the Roman saints...But from a manuscript of Torrigio's we see that on the sarcophagus in question there were other letters beside the word Linus, so that they rather belonged to some other name (such as Aquilinus, Anullinus). The place of the discovery of the tomb is a proof that it could not be the tomb of Linus (De Rossi, "Inscriptiones christianae urbis Romae", II, 23-7) (Kirsch J.P. Transcribed by Gerard Haffner. Pope St. Linus. The Catholic Encyclopedia, Volume IX. Copyright © 1910 by Robert Appleton Company. Online Edition Copyright © 2003 by K. Knight. Nihil Obstat, October 1, 1910. Remy Lafort, Censor. Imprimatur. +John M. Farley, Archbishop of New York).
So you see, the entire pyramid comes crashing down and was based on supposition and apocryphal writings. There is no such thing as apostolic succession. So speculation on how anyone feels about it or believes about it is immaterial.
You didn't read it correctly. The Methodist author is criticizing that approach just the same way you are. The author is agreeing that that way of understanding apostolic succession has historical problems. You are also saying that that way of understanding apostolic succession has historical problems.
Please read it again, because I think you missed his further point on the correct way to think about apostolic succession.