My intention is not to make a theological critique of Peter as pope but rather a historical one. Having said that, I am a protestant Christian and so I do have some theological opinions on the matter. I am just putting my bias on the table.
Roman Catholics and protestants debate the meaning of Matthew 16:18.
And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it.
I actually do not see the question of whether the rock is Peter or the confession he gave in the preceding verses as being relevant to the question at hand.
Even if Peter the individual was the rock on which Jesus would build his church, that does not mean that Peter was the first of many popes.
Of course, there was nothing like the modern idea of a pope then or for many years after this. The most we could discuss would be if Peter was the first bishop of Rome, as that is what a pope really is, the bishop of Rome.
I believe there are good traditions placing both Peter and Paul in Rome at some point and that they were both martyred in that city. But was Peter considered to be the bishop of Rome? No doubt, Peter was considered a man of authority, being one of the pillars of the church (along with James and John). See my podcast episode on the Pillars.
But that does not mean that Peter was the bishop of Rome. That type of idea does not appear to some time later. We start to see this thought developing with Irenaeus. Two passages are relevant.
[T]he very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. …
The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate.
You can read these passages in context here.
When it comes to Irenaeus’s claim that Paul and Peter founded the church at Rome, he is simply wrong. It is clear at the end of Acts, when Paul arrives in Rome that there is already a Christian community. When Paul wrote his letter to the Romans, he had not yet visited Rome. Nor is there any indication that Peter was already there.
Most scholars believe that the Roman church began when Jewish visitors from Rome became followers of Jesus on the day of Pentecost in Jerusalem (Acts 2). They brought back their faith and the Roman church began as predominantly Jewish, only later adding a Gentile component.
What role Peter had at Rome cannot be said. There may have been someone in the role of a bishop before he got there. There is no contemporary evidence that Peter was a bishop. His role as an apostle, which generally was based on the itinerant model rather than the settled bishop model, may have been enough.
Did Peter and Paul appoint Linus as bishop of Rome? Perhaps. We cannot know. But there is no contemporary or even near-contemporary evidence that special authority was being passed on from Peter to Linus. Surely Peter had appointed many leaders over his ministry. Why would Linus have a special role?
This is not meant to be an attack on the papal system. But we should be aware that the concept of Peter as the first pope is a theological statement and not one based on good historical evidence. People are welcome to accept this on faith if they so choose.