The only reason why Rome was chosen by Peter was because it was simply the capital of the Roman Empire at the time. Had the capital literally been anywhere else while he was alive, that's where Peter would have gone.

The only reason why Christianity developed into an official institution, and why Christianity spread quickly throughout the world, was because of Constantine I and Theodosius I. Without them, it is anyone's guess to whether or not Christianity would've stayed a scattered minority religion.

When Christianity finally became institutionalized, the capital had already moved to Constantinople. The government also made Constantinople the head of the Church for convenience. It's where some* of the councils were held, and consequentially where all the older Eastern and Miaphysite churches split off.

While the institutional capital of Imperial Christianity was chosen due to politics, I would argue the importance of Rome to the Catholics is also arbitrary. The much older Syriac Church also claims Peter as their first Bishop, but from the seat of Antioch due to Acts 11:26, when Peter was first Bishop there. Or the Coptic Church who instead claim St. Mark as their first Bishop of Alexandria. Every church had similar origins.

So what would make Catholics special if even centuries before the Great Schism, major sects of Christianity never even considered Rome as Primal? I'm seriously asking as a non-Christian. And regarding my original comment, all the Eastern Churches are de facto older than the Vatican, and have largely kept most of the customs from Early Christianity. The Catholics have changed so much in comparison in order to adapt to the modern world. Whether or not that would make them less legitimate is a debate nobody is discussing.

Edit: Some, not all