Why is the Bishop of Rome the Pope?

Yes, why is the Bishop of Rome the leader of Jesus' Church on earth--or Pope? There are two distinct issues here. One is about Petrine leadership, and the second issue is why the Bishop of Rome is Peter's successor.

I have four questions. 

1. Did Jesus make Peter leader of his disciples and Church?
2. Does the Church need this Petrine leadership?
3. Is Petrine leadership transferable?
4. Why is Rome the instantiation of Petrine leadership?

Did Jesus make Peter leader of his disciples and Church?
This is an easy one. Yes!  John 21:15-19 is the clearest expression of this. Peter loved Jesus more than any of the other disciples, and Jesus told him to tend his lambs, feed his sheep, and feed his sheep. This is unambiguous. 

Does the Church need Petrine Leadership?
Yes!
Mt 16:18 tells us that Jesus declared to Cephas, "You are Peter" and "upon this rock I will build my church . . ." This follows Peter confessing that Jesus is the Messiah and the Son of God. And Jesus says that God revealed that to Peter. It was not a deduction, but it was a gift of faith that no one else possessed (except, of course, Mary and maybe his family). The "this" of Mat 16:19 may be Peter or it may be Peter's declaration, but it doesn't matter. Because the Church is built on the testimony of the Apostles and not abstract claims. So, even if one argues that the "this rock" does not refer to Peter, it refers to Peter's confession (not some abstract claim). His confession that Jesus is Messiah and the Son of God was revealed to him, and it is his faith that becomes foundational for what is to follow. The Christian faith is not based on the fact that Jesus Christ is the Son of God and Messiah, but that Peter and the other apostles say that Jesus is the Son of God and Messiah. The faith is 100% based on the testimony of the Apostles, especially Peter.


So the question is not whether Peter's leadership is necessary. That's irrelevant. The issue is that Jesus declared it to be so. 

Is Petrine Leadership Transferable?
Now this is where it gets tricky and strange. I don't believe that the faith demands it. However, two things are relevant here. One is that both Matthew and John indicate that Peter's leadership is ongoing. In John, as long as there are lambs and sheep, then Peter's leadership is what Jesus desires. In Matthew, as long as there is a church being built, then the rock is needed. So it would stand to reason that as long as there's a church, Petrine leadership is needed and, unless Peter were to live forever, someone else would need to continue that leadership. The other piece of this discussion, which is underrated, is that the Church has the power to institute the transferability of Petrine leadership. Mt 16:19 says, "I will give you the keys to the kingdom of heaven. Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven." The Church has the authority to continue Peter's leadership, whether it is explicitly declared or not. The issue is did the Church do so? The answer has to be yes, since we have had Petrine leadership for centuries now. So my take is that Jesus didn't authorize any transferability of Petrine leadership, but the Church can and did.

Why is the Bishop of Rome the Successor of Peter?
 . . .
The arguments are that Peter was Bishop of Rome (such as it is) and that both Peter and Paul were martyred in Rome, making Rome special. The latter argument is ridiculous. I'll up you one. Jesus died in Jerusalem. Thus, by the logic where deaths have occurred, the Bishop of Jerusalem should then be the successor of Peter and the primary see. 
Can the Bishop of Rome validly claim to be Peter's successor? Yes, and the Roman church did, and it was unchallenged for the most part and accepted. And by Matthew 16:19 logic, the Church bound this in heaven and earth. However, Rome has no necessary claim to Petrine leadership. Just as the Church allowed Rome to claim that mantle, it can remove it. Although, practically, it would take an extreme emergency to agree that the Bishop of Rome no longer has an intrinsic claim to the papacy.

So . . . Petrine leadership is real, is continuous, it's transferable, and its intrinsic link to Rome all check the necessary boxes.