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Perhaps more to the point, none of his disciples listening that day, in any language, understood it, since they debated a short time later who was the greatest.

And when they did, Jesus didn’t reply, “Peter”. Instead, he made it plain the entire argument was wrong.

And since all the disciples were granted the power to bind and loose a couple of chapters later, I find it hard to believe that Peter was made Vicar of Christ. If Jesus did, he sure hid it well.

Well, the fact you bring up, that the disciples did argue, makes MY point, not yours!
Though all of Scripture is Sacred, we must understand that even the first 12 did not agree, and that Paul did not always agree with the remaining 11.

So, Scripture is the work of man, inspired by God, but even Scripture shows us differences of opinion, among good, solid, Saintly Christians.

Also, Peter was head of the Church, on Earth, AFTER Crucifixion and Resurrection.

Jesus appointed his Successor on Earth, before Jesus left this Earth.

Peter did the same thing.