In Luke 22:31–32, Jesus says:

“Simon, Simon, behold, Satan demanded to sift you all like wheat; but I have prayed for you (singular), that your faith may not fail; and when you have turned again, strengthen your brothers.”

Jesus prays for all the apostles, but intercedes individually for Peter, giving him the mission to strengthen the rest.
That’s why early Christians saw Peter’s office as one of unity and confirmation.

The bishop of Rome, as Peter’s successor, continues that same ministry, the visible center of unity Christ intended for His Church.

A movement without a singular leader is bound to fragment and fail ultimately.