1. Your understanding is not correct. The rock is the revelation that Jesus is the Son of God. This is the cornerstone of the church.
2. We never see Peter as the leader of the church in the rest of scripture.
3. In fact, during the Jerusalem Council meeting with Peter present (Acts 15), James the brother of Jesus presides over the council, not Peter.
If Christ had meant to say: "The Rock is the Revelation, and on that Revelation, I will build My Church", He would have done so. But He didn't. Here's another (Lutheran) commentary: "The obvious pun which has made its way into the Gk. text as well suggests a material identity between
petra and
petros, the more so as it is impossible to differentiate strictly between the meanings of the two words. On the other hand, only the fairly assured Aramaic original of the saying enables us to assert with confidence the formal and material identity between
petra and
petros:
petra =
Kepha =
petros....Since Peter, the rock of the Church, is thus given by Christ Himself, the master of the house (Is. 22:22; Rev. 3:7), the keys of the kingdom of heaven, he is the human mediator of the resurrection, and he has the task of admitting the people of God into the kingdom of the resurrection...The idea of the Reformers that He is referring to the faith of Peter is quite inconceivable in view of the probably different setting of the story...For there is no reference here to the faith of Peter. Rather, the parallelism of 'thou art Rock' and 'on this rock I will build' shows that the second rock can only be the same as the first. It is thus evident that Jesus is referring to Peter, to whom He has given the name Rock. He appoints Peter, the impulsive, enthusiastic, but not persevering man in the circle, to be the foundation of His
ecclesia. To this extent Roman Catholic exegesis is right and all Protestant attempts to evade this interpretation are to be rejected." (Cullmann, article on "Rock" (
petros, petra) trans. and ed. by Geoffrey W. Bromiley,
Theological Dictionary of the New Testament [Eerdmans Publishing, 1968], volume 6, page 98, 107, 108)"
Regarding your second point, no. "There is ample evidence in the New Testament that
Peter was first in authority among the apostles. Whenever they were named, Peter headed the list (Matt. 10:1-4, Mark 3:16-19, Luke 6:14-16, Acts 1:13); sometimes the apostles were referred to as “Peter and those who were with him” (Luke 9:32). Peter was the one who generally spoke for the apostles (Matt. 18:21, Mark 8:29, Luke 12:41, John 6:68-69), and he figured in many of the most dramatic scenes (Matt. 14:28-32, 17:24-27; Mark 10:23-28). On Pentecost it was Peter who first preached to the crowds (Acts 2:14-40), and he worked the first healing in the Church age (Acts 3:6-7).
It is Peter’s faith that will strengthen his brethren (Luke 22:32) and Peter is given Christ’s flock to shepherd (John 21:17). An angel was sent to announce the resurrection to Peter (Mark 16:7), and the risen Christ appeared first to Peter (Luke 24:34). He headed the meeting that elected Matthias to replace Judas (Acts 1:13-26), and he received the first converts (Acts 2:41). He inflicted the first punishment (Acts 5:1-11) and excommunicated the first heretic (Acts 8:18-23). He led the first council in Jerusalem (Acts 15) and announced the first dogmatic decision (Acts 15:7-11). It was to Peter that the revelation came that Gentiles were to be baptized and accepted as Christians (Acts 10:46-48)."
https://www.catholic.com/tract/peter-and-the-papacy
Regarding your third point, St. James was First Bishop of Jerusalem, where that Acts 15 Council was held. St. Peter, as history shows, was First Bishop of Rome. Even in spite of that, in that Jerusalem Council, Peter spoke first, in vs 7-11, and James after that, in vs 14-21.
Protestant Historian Philip Schaff, who wrote a scholarly work, "History of the Christian Church" (on Amazon here:
https://www.amazon.in/History-Christian-Church-8/dp/156563196X ) gives the list of the First Four Bishops of Rome from St. Peter.
"The
succession list of bishops in the apostolic see of Rome of the first two centuries as provided by Schaff (volume 2, page 166) is --
- St. Peter (d. 64 or 67)
- St. Linus (67-76)
- St. Anacletus (76-88)
- St. Clement I (88-97) ...
On St. Clement of Rome (c. 96 AD), reckoned as the fourth Pope from St. Peter, Schaff states "...it can hardly be denied that the document [Clement to the Corinthians] reveals the sense of a certain superiority over all ordinary congregations. The Roman church here, without being asked (as far as appears), gives advice, with superior administrative wisdom, to an important church in the East, dispatches messengers to her, and exhorts her to order and unity in a tone of calm dignity and authority, as the organ of God and the Holy Spirit. This is all the more surprising if St. John, as is probable, was then still living in Ephesus, which was nearer to Corinth than Rome." (Schaff, volume 2, page 158) and "It must in justice be admitted, however, that the list of Roman bishops has by far the preminence in age, completeness, integrity of succession, consistency of doctrine and policy, above every similar catalogue, not excepting those of Jerusalem, Antioch, Alexandria, and Constantinople...." (Schaff, page 166)
In spite of this, if you believe, Christ setup His Church without any Hierarchy and Primacy among Bishops or Pastors, fine. I believe Christ clearly intended St. Peter to be Chief Bishop or Lead Pastor among the Apostles, something which the Early Church, right from the First Centuries discernibly kept, and something which the Catholic Church retains till today. God Bless.