Hello, Hope.
While Matthew 16:18 is normally cited these days as 'proof' of Christ making Peter the head of his church, ordinary tradition doesn't support it. The Early Church Fathers, often referred to as authoritative by the RCC, are all over the place in their comments about who, if anyone, might be some kind of natural "leader" in the church.
Some said Peter, but others said Peter and Paul jointly. Still others said John, or James, or even three of the above-mentioned jointly. And the bishops of Rome themselves did not cite Matthew 16:18 as evidence for their claims until 300 or so years later, and the very idea of a Papacy did not emerge until about that time.
If we turn from the ECF's to the church in general, there is evidence of the bishop of Rome being respected from the mid-first century onward...but for non theological reasons. The diocese of Rome was the largest and wealthiest of all Christian dioceses at that time; and the city of Rome (the "Eternal City") and seat of government in the Empire was of course more influential than other Christian centers. Naturally, the bishop had influence for those reasons.
If we really insist upon tradition deciding the matter, all of the above notwithstanding, then what do we say about the Eastern churches which never, ever agreed to any one bishop being the head of the universal church? We are accustomed to the Orthodox Eastern churches denouncing that view, but those successors of the Apostles always did!
So by the RCC's own definition of tradition, the Papal claims are refuted.
In addition, even if one were persuaded to think that Peter was made some sort of Pope by Christ, nothing whatsoever is said about successors taking over after Peter's death and passing the role on to their own successors! That, as we might expect, is never discussed by the Church, but just left to be assumed. Apostolic Succession is supported in various ways, but the passing on of a Papal role is not. In fact, the immediate successor to Peter as bishop of Rome, St. Linus, was not even in Rome at the time and had not been named as successor or consecrated by Peter. The townspeople liked him and simply installed him on their own.