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The N/T is rife with references to St. Peter's ecclesiastical authority, even in those places where it's a secondary reference. Peter was the only apostle who was allowed by Christ to walk on water. When Peter and John ran to the empty tomb of Christ John arrived first, "but did not enter in"; he waited for Peter who entered first. Peter and his brother were selected by Jesus to be apostles, the brother was older making him first in Jewish law and tradition... however, Peter was selected first over his elder brother. The N/T often refers to the apostles as "Peter and the eleven". It was Peter who rose up and silenced the debate at the first Christian council in Jerusalem. Then there are more direct references which you have already mentioned.

Still, Protestants will protest, (no pun intended), about the successors to Peter being given the same authority. But apostolic succession is also clear in the bible, such as when Judas apostacized he had to be replaced by another. There is also the laying on of hands to ordain others, etc. As you said, Peter's authority is not 'absolute' because it must always be perfectly aligned with what Jesus Christ taught. None-the-less, Peter's, (papal), word regarding the faith is authoratative and final. His infallibility, when speaking ex cathedra regarding matters of faith and morals, is protected by the Holy Spirit from error. Without a vicar of Christ to shepherd the faithful the Catholic Church would become divided as are the hundreds of Protestant churches with differing, opposing theology and doctrines.