It explains that Peter came to Rome first, and it describes how Paul eventually joined Peter. But the narrative begins with Peter already being in Rome.
but all the apostles were given the same powers as he had, binding and loosing. /
Not exactly. The Keys were given to Peter only, and the power to bind and loose was given him first. Two chapters down, in Matthew 18 the power to bind and loose is given to the apostles also, in the context of the church legislation. It is reasonable to conclude that Peter's binding and loosing referred to the matters of salvation (as he holds the keys to the Kingdom of heaven), as the Apostles deal solely with Church matters.
Only Peter was charged with the feeding and shepherding the sheep.
Jesus certainly meant His Church to be till He comes again, even as He foresaw the fracturing. This is explicit in the promise of gates of Hell not prevailing, in the image fo the Innkeeper in the parable fo the Good Samaritan, and in His prayer in John 17 "that they may be made perfect in one: and the world may know that thou hast sent me, and hast loved them, as thou hast also loved me. Father, I will that where I am, they also whom thou hast given me may be with me; that they may see my glory which thou hast given me, because thou hast loved me before the creation of the world." Peter also understood his "tabernacle" to extend in perpetuity: "I will endeavour, that you frequently have after my decease, whereby you may keep a memory of these things" (2 Peter 1).
James was the presider over the first council in Jerusalem and spoke with authority there, not Peter.
This is because James was the head of Jerusalem Church. Nevertheless you see that Peter speaks first and sets things in motion, while James concludes the proceedings.
So which is it, church or churches?
I think that when we read of churches, these are local organizations headed up by a bishop. It is clear that they are all to follow the same doctrine, as the Letters put back into line those that strayed. Again, Christ prayed that they be one and on other occasions He spoke of His Church as a single unit.
Church of Rome ended up prevailing in Western history
As a visible symbol of unity, that could be nowhere else but in imperial Rome. It is however, incorrect to say that the local churches operated independently as the local heresies were put out by acts of ecumenical councils. It is true that the early Church was far more consiliar than the later, more authocratic Latin model of the middle ages and till today. But on the other hand, do not forget that the church today is primarily the Latin Church and the Pope's strong hand is as a patriarch of the Latin Church. The Churches of the East, for example, are pretty much left to their own devices (they even don't insert the Filioque)because they face no heresies and are very traditional, while all the heresy in the past 5 centuries is coming from the Latin West.
The earliest example of a Pope exercising authority across the head of the local bishop is Pope Clement I reaching to Corinth to demand reinstallment of certain bishops over the head of no less a figure than apostle John! (Letter to the Corinthians).
nothing about a rock or cornerstone or one foundation having eminence over the others there, it sounds equal.
As it will be, as it was at the table of the Last Supper. The New Jerusalem is Christ in glory and His bride unified and in glory. At that point, the Papacy has filfilled its centrifying role.
I'll ponder what you wrote in #128. Had something else in response, but flushed it, could be taken wrong. That took some effort to put that together, and I thank you for it.
Would you know if Jerome included this apocryphal work in his Latin Vulgate?. And whether it ever appeared on the list of the early church known as "non-recipiendi", ie, "not to be received [believed]". Wasn't this well known even by Jerome to be a heretical work of Gnostic fiction? Certainly this discredited work was not relied upon as the source of the great legend of that 25 year Petrine Bishopric in Rome or was it?
The earliest example of a Pope exercising authority across the head of the local bishop is Pope Clement I reaching to Corinth to demand reinstallment of certain bishops over the head of no less a figure than apostle John! (Letter to the Corinthians).
It appears hardly to be a "demand" leter. Rather a pastoral letter from one Church to another.