The CC can claim all it wants to. Lets look for facts to back these claims up. Again, where do we see Peter ruling from Rome? Where in Scripture do we see the other apostles supporting the idea that Peter is the leader of the entire church? In all their writing not one makes such a claim about him like this.
You are allowing yourself to be contaminated by a secular notion of leadership. Jesus was very clear that the one who wanted to be the leader must be the servant of all. Peter is understood this teaching, and took on the attitude of a servant, not a “ruler”.
Peter does have primacy throughout the scriptures.
Not if you can’t support it with facts or Scripture.
Didn’t you ever notice that his name appears first in all the lists?
Primacy of Peter:
Matthew, chapter 16
18: And I tell you, you are Peter, and on this rock I will build my church, and the powers of death shall not prevail against it. 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."
Luke, chapter 22 – Jesus tells Peter to strengthen his brethren
31: “Simon, Simon, behold, Satan demanded to have you, that he might sift you like wheat,
32: but I have prayed for you that your faith may not fail; and when you have turned again, strengthen your brethren.”
John, chapter 21 – Peter is given Christ’s flock to shepherd
17: He said to him the third time, “Simon, son of John, do you love me?” Peter was grieved because he said to him the third time, “Do you love me?” And he said to him, “Lord, you know everything; you know that I love you.” Jesus said to him, "Feed my sheep.
Mark, chapter 16 – Peter is mentioned separate from the others
7: But go, tell his disciples and Peter that he is going before you to Galilee; there you will see him, as he told you."
Luke, chapter 24 – risen Jesus appeared to Peter first
34: who said, “The Lord has risen indeed, and has appeared to Simon!”
Acts, chapter 1 – Peter headed the meeting to elect Matthias
13: and when they had entered, they went up to the upper room, where they were staying, Peter and John and James and Andrew, Philip and Thomas, Bartholomew and Matthew, James the son of Alphaeus and Simon the Zealot and Judas the son of James.
14: All these with one accord devoted themselves to prayer, together with the women and Mary the mother of Jesus, and with his brothers.
15: In those days **Peter stood up among the brethren **(the company of persons was in all about a hundred and twenty), and said,
16: "Brethren, the scripture had to be fulfilled, which the Holy Spirit spoke beforehand by the mouth of David, concerning Judas who was guide to those who arrested Jesus.
Acts, chapter 2 - Peter leads the Apostles
14: But **Peter, standing with the eleven, lifted up his voice **and addressed them, "Men of Judea and all who dwell in Jerusalem, let this be known to you, and give ear to my words…
37: Now when they heard this they were cut to the heart, and said to Peter and the rest of the apostles, “Brethren, what shall we do?”
38: And Peter said to them, “Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit.
39: For the promise is to you and to your children and to all that are far off, every one whom the Lord our God calls to him.”
40: And he testified with many other words and exhorted them, saying, “Save yourselves from this crooked generation.”
41: So those who received his word were baptized, and there were added that day about three thousand souls.
Acts, chapter 3 - Peter performs first miracle after Pentecost
6: But Peter said, “I have no silver and gold, but I give you what I have; in the name of Jesus Christ of Nazareth, walk.”
7: And he took him by the right hand and raised him up; and immediately his feet and ankles were made strong.
Acts, chapter 5 – Peter inflicted first punishment
1: But a man named Anani’as with his wife Sapphi’ra sold a piece of property,
2: and with his wife’s knowledge he kept back some of the proceeds, and brought only a part and laid it at the apostles’ feet.
3: But Peter said, “Anani’as, why has Satan filled your heart to lie to the Holy Spirit and to keep back part of the proceeds of the land?
4: While it remained unsold, did it not remain your own? And after it was sold, was it not at your disposal? How is it that you have contrived this deed in your heart? You have not lied to men but to God.”
5: When Anani’as heard these words, he fell down and died. And great fear came upon all who heard of it.
6: The young men rose and wrapped him up and carried him out and buried him.
7: After an interval of about three hours his wife came in, not knowing what had happened.
8: And Peter said to her, “Tell me whether you sold the land for so much.” And she said, “Yes, for so much.”
9: But Peter said to her, “How is it that you have agreed together to tempt the Spirit of the Lord? Hark, the feet of those that have buried your husband are at the door, and they will carry you out.”
10: Immediately she fell down at his feet and died. When the young men came in they found her dead, and they carried her out and buried her beside her husband.
11: And great fear came upon the whole church, and upon all who heard of these things
Acts, chapter 8 – Peter is first to excommunicate a heretic
20: But Peter said to him, "Your silver perish with you, because you thought you could obtain the gift of God with money!
21: You have neither part nor lot in this matter, for your heart is not right before God.
Acts, chapter 10 Peter receives divine revelation
44: While Peter was still saying this, the Holy Spirit fell on all who heard the word.
45: And the believers from among the circumcised who came with Peter were amazed, because the gift of the Holy Spirit had been poured out even on the Gentiles.
46: For they heard them speaking in tongues and extolling God. Then Peter declared,
47: “Can any one forbid water for baptizing these people who have received the Holy Spirit just as we have?”
48: And he commanded them to be baptized in the name of Jesus Christ. Then they asked him to remain for some days.
Acts, chapter 15 – Peter led the first council in Jerusalem
7: And after there had been much debate, Peter rose and said to them, "
Brethren, you know that in the early days God made choice among you, that by my mouth the Gentiles should hear the word of the gospel and believe.
Galatians, chapter 1 – Paul visits Peter after conversion
18: Then after three years I went up to Jerusalem to visit Cephas, and remained with him fifteen days
In several other places Peter’s name is mentioned separate from the others as well as speaking for them. If you understand anything abut the writings of that time you’ll understand why. Also, Peters name is mentioned 195 times, more than all the other apostles combined.