Non-Catholics and anti-Catholics often tell us that Rome wasn’t that important, that there were other patriarchates like Alexandria, Jerusalem, and Antioch, and that the ascendancy of Rome took place at some later time, in other words, was not apostolic in origin. An understanding of ancient hierarchies will show that Rome was, indeed, primary in ancient times, and that because of Peter’s importance, with the church in Rome founded by Peter, Prime City + Prime apostle = Church of Rome.
There is an unofficial formula for determining the importance of a bishop in the grand scheme. It is filtered today because of how far-flung the Church is, but when it all boils down to it, a bishop’s importance in ancient times was determined by a)how important the city or town was + b) how important the bishop’s ancestor, the apostle or saint who established the local church is. Example: Jerusalem was an important crossroads in the Roman empire. The traditional founder of the Patriarchy, James is very important, being second in the traditional list of apostles. Therefore, the Patriarchate of Jerusalem was very important. Alexandria was another very important city in the Roman empire, the church there was founded by St. Mark. Mark, a very important figure, in a very important city, makes the church in Alexandria very important. Of course we have the city where all roads lead, Rome, being established by the prime apostle, Peter, so from the very founding of the church in Rome, This was the primary city of the Catholic Church. So when the focus of the secular world switched to Constantinople, and a patriarchate was founded there, why didn’t the Church’s focus switch there as well? Constantinople was THE important city now, because Rome was sacked and overrun, and Jerusalem had been destroyed, eventually Muslims overran Alexandria and all of north Africa. So why didn’t the church make their new home in Constantinople? Who founded the Church in Constantinople??? Why Constantine himself. In Catholic eyes, while Constantine did great things to forward the Church’s cause, making it legal for Christians to be Christians, he was no saint, and certainly no apostle. He installed the bishop, and engineered the growth in status to that of patriarchate. But Constantine was a secular ruler, not an apostle and not a saint, so Constantinople, by all accounts was never of the same status as Rome.
So the reason that Rome is the center of the Catholic Church is because of the status of Rome in the ancient world, and the status of Peter as primary of all the apostles.