A leader, yes. THE leader, no
The Keys and the Feed my Sheep episode appoint the single leader, and it is indicated by his leadership in winning the first converts, leading in the issue of dietetic law, and several miracles described in Acts. St John Chrysostom called Peter Prince of the Apostles. It is true that St, Paul lists also St. James and St. John as "pillars".
The Rock was Peter's bold confession about Jesus
This is your opinion? The renaming indicates a personal attribute of Simon bar Jonas. You can spin it that way, -- I have no serious objection to it, as indeed his confession prompted the renaming and the keys episode, but it is your commentary, nothing more.
Jesus gave the Keys to the church itself (as is seen just two short chapters later), not Peter exclusively
This is not accurate scripturally. The Apostles are given the power to bind and loose in the context of the teaching authority of the Church, but the keys are not given them.
Reread the text. The keys are the power to bind and loose. In Matthew 18, he gives this same power to the church (not the centralized organization, but the body of all believers). Verse 1 speaks of the "disciples" not the "apostles" per se.
Then, IN CONTEXT: Jesus says later in the same chapter,(Matt 18)
15Moreover if thy brother shall trespass against thee, go and tell him his fault between thee and him alone: if he shall hear thee, thou hast gained thy brother.
16But if he will not hear thee, then take with thee one or two more, that in the mouth of two or three witnesses every word may be established.
17And if he shall neglect to hear them, tell it unto the church: but if he neglect to hear the church, let him be unto thee as an heathen man and a publican.
18Verily I say unto you, Whatsoever ye shall bind on earth shall be bound in heaven: and whatsoever ye shall loose on earth shall be loosed in heaven.
19Again I say unto you, That if two of you shall agree on earth as touching any thing that they shall ask, it shall be done for them of my Father which is in heaven.
20For where two or three are gathered together in my name, there am I in the midst of them.
Peter then asks a question regarding forgiveness and Jesus starts telling him about forgiveness and how it is in the Kingdom of Heaven.
Similarly, in the Keys of the Kingdom passage (Matt 16) Jesus says: 19And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.
Again, when the RISEN Lord showed himself to the DISCIPLES that had gathered He said:
John 20:23Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.
This was BEFORE there was an organized church anywhere. After this, the church spread like wildfire. Paul travelled most extensively, but Peter's words seem addressed not to the Romans but to those in Asia Minor. It can not be proven that Peter was ever in Rome and his epistles, for someone who is supposedly the 1st Pope are somewhat narrow in focus. They mainly deal with encouragement to the persecuted church and wasn't a doctrinal letter per se like Paul's letters were.
Again, if he were the first Pope, why not mention it? Even hint it? No special exclusive power was given to Peter to do anything. And Peter is NEVER EVER referred to as the head of the church on earth. He was one of the Lord's great apostles. And JESUS Himself is head over the Church.