Tertullian (~200 AD) calls him Peter's successor, also internal evidence within his letter has lead many (including then Cardinal Joseph Ratzinger) to conclude it was written before the destruction of the Temple.
Well the Temple was destroyed in 70AD, and Tertullian wasn't born until 155AD, so I don't see how he could have written a letter before he was born.
As far as the phrasing, "Peter's successor" has been used to refer to many different Popes over the years, including the current Pope Leo XIV. It refers to the fact that the Pope is the current successor to the line of Saint Peter, not just specifically his immediate successor in that line.
Can you tell me more about this decision making?
First let me preface the rest of this by saying that the Papal Line and the history of Papal Succession is not my area of expertise, I am merely going off my understanding of it. It's possible for me to be mistaken to varying degrees and if someone who is more well versed on the matter wants to chime in I welcome it.
the Catholic way of this was a very specific organization through which the power given to Peter by Jesus has continued to this day, I would be more inclined to pick the Catholic viewpoint if I could see more clear evidence exactly how Peter's role is leader clearly and distinctly transferred to the next guy.
I think it's important here to clarify the depth of the Catholic position, and I will do my best to show it's basis in scripture. We can all agree that Christ is not just King, but the King of Kings. This means that he has a Kingdom, and unless you believe something very different than I do, that Kingdom includes all of existence, which itself includes Earth. While a King is the highest authority in a Kingdom, he rarely administers it alone. Typically it is impossible for a given King to run an entire Kingdom without help, and while Christ is arguable the exception to that rule, it makes sense that he would follow a traditional structure not because he has to but for our benefit as his subjects. We see this in the Gospels, Christ could have gone out to all the people personally, but instead he sent his Apostles.
So we have a King and we have a Kingdom, and for the sake of the subjects that Kingdom needs administrators. The top administrator of a Kingdom is traditionally called the Steward, and the symbol of the office of Steward are the keys to the Kingdom. We see this in the Kingdom of David per Isaiah 22:15-25, where the corrupt steward Shebna is struck down and replaced with the righteous Eliakim. With specific attention to Isaiah 22:22 we see the significance of the keys and the powers that come with them.
The verse reads:
22 And I will place on his (Eliakim's) shoulder the key of the house of David; he shall open, and none shall shut; and he shall shut, and none shall open.
Now we can draw a direct parallel to Matthew 16 when Christ ordains Peter as the rock of the Church and grants him the keys and powers.
The verse reads:
19 I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”
If we agree that Christ is our King, and the Earth is his Kingdom, then it lines up that Peter is his Steward. In the structure of a Kingdom, the only authority higher than the Steward is the King, so when the King is away from the Kingdom, the Steward is the highest authority. So when the Steward is faced with the reality of his own mortality, and knows the importance of his office, he has the authority to transfer that power to a new Steward, or establish a system by which a new Steward is chosen. The best evidence for this happening is that we continue to have Stewards with a direct line traced back to the first one.
You mentioned Paul ordained Linus but certainly he and Peter ordained more guys, Clement for example was consecrated by Peter.
They ordained him once as a what we would call a Priest or perhaps a Bishop today, but when someone becomes Pope it is an additional ordination. So I meant that Paul ordained him as Pope specifically. Granted it the office wasn't called "Pope" then. However as we see with Tertullian the office was still referred to as Peter's successor.
Was it by virtue of being the highest ranking man (I'm not sure how that would be determined) in the city where the previous Pope died, did it have some special relation to the city of Rome?
Well the office we know as Pope has long been considered the first among equals, so a traditional system of rank doesn't exactly fit in. When I referred to Peter and Paul as the two top apostles I meant so more in terms of importance than a traditional ranking system. As the Church grew and developed a system of rank became necessary, but back then there wasn't a need for delineation between Priests and Bishops. Especially in the very early days, all 12 Apostles would be like Bishops are now.
Did Peter and the other subsequent early Popes, under the threat of persecution and martyrdom make known their choice of successor just in case?
Such a practice would make logical sense, but I suspect they didn't simply name a successor so much as confer with their immediate peers and reach a consensus as to who is best suited to lead next. There is certainly more indication of a collaborative decision process than Popes unilaterally declaring a successor.
If I could see good clear evidence of an early successor system meant to transfer Peter's power to the next guy worked out during Peter's time then certainly I'd be hard pressed to say the Church Jesus referred to wasn't the Roman Catholic Church.
Honestly IMO some of the best evidence we have is that there has been a traceable line of successors to this day. It is not as if they were conjured from nothing afterall. The Great Schism didn't occur until 1054 so even if you ask the Orthodox the line of Popes from Peter to Leo IX is recognized. The first protestants didn't split off for another 500 years after that.
Anyway, I have said this is not my area of expertise so the best I can do is show how it is rooted in scripture, how it makes logical sense, and how it is evident by history, which is what I have tried to do here.