This isn't a question that can be satisfactorily answered by modern scholars and is going to invariably run into issues of religious devotion, or lack thereof. It is all going to come down to how much stock you place in Church traditions and later accounts, as there is quite simply nothing else to base it on.

The Roman Catholic Church claims to be the Universal (catholic) Church as implemented by St. Peter, the eponymous Rock upon which Christ's church was to be built (Matthew 16:18) Now unfortunately our literary record for the earliest Christian church is sparse. By long Catholic Church tradition, St. Peter is held to have been the first bishop of Rome (and by extension Pope) and there is nothing certainly to contradict this, and it was widely believed by the time there was a vibrant literary culture in Christianity, but this still sidesteps the main issue, the lack of contemporary written sources, whether narratives, inscriptions, or whatnot. We don't simply have the sources to determine the veracity of these sorts of ideas.

All of our accounts of St. Peter's life for example come from scripture or from scripture adjacent writings (many of dubious accuracy) and indeed for much of the first three centuries of its existence as an organized religion Christianity is rather obscure. The first concrete records of Christians by outside sources, such as Josephus and Tacitus describe events during the later reign of Nero, when Christians were blamed by the emperor for starting the Great Fire of Rome. That occurred in 64 AD and Peter's martyrdom is ascribed to the same time frame...by the Catholic Church (and other Churches as well such as the Orthodox). This is going to be a running problem. There are quite simply not enough surviving documents to establish a firm line of causality. There aren't even contemporary documents that place Peter unambiguously in Rome by that time. The first firm documentary evidence of Roman Christians comes from around the turn of the 1-2nd centuries and the letters from Pope Clement I to the Corinthians, by which time many of the ideas around Peter had started to solidify.

It is not until the 3rd century that there are enough surviving documents to really reconstruct a timeline of the early Church and its leaders. Certainly by that point the belief that the Roman Church had been founded by Peter who had been martyred there was well established, and the historicity of figures like Jesus and Peter is not seriously doubted by most scholars today. The Catholic Church claims its authority from the principle of "Apostolic Succession" by which the power and authority invested into the apostles by Christ was handed down to the apostles and then from them to the members of the clergy, and in the total absence of literary, archaeological, or documentary evidence it remains a matter of faith for individuals to parse.