Ralph, that is a great question. I’m glad you asked.
I’m sure you agree that Jesus’ Apostles were important fellows and that history shows that they all went off to preach, get converts and start Churches.
I wish I knew everything, but I don’t. I do know that St. Thomas went to India and started the Church there. I do know that St. Mark went to Alexandria and started the Church there. St. Philip in Ethiopia, St. Andrew went to Byzantium and started a Church there, St. Peter went to Antioch and started a Church there, St. Paul went all over Greece: Corinth, Thessalonica, etc and started Churches there, Sts. Peter and Paul went to Rome and started a Church there, St. James remained in Jeruselem and started the Church there. I wish I knew where the others went to start Churches. I know that John was the caretaker of the Theotokos.
Anyway, so what made some of these Churches “more important” that they would be raised to called Patriarches and the others weren’t? Well, it has to do with their importance in the secular world and in the previous Jewish faith.
Obviously, Jeruselem was important because that is where the Temple where the right worship of the Triune God was done in the Pre-Christian world.
Okay, so what about the others?
Well, Rome was the Capitol City of the known world…the Roman Empire which had ruled for centuries before and after Christ’s death and resurrection.
Although the Church existed, as established by St. Andrew in 38ad, in Byzantium, it was not elevated to be called “Patriarch” until after the Roman Emperor Constantine transferred the Roman Empiror’s Capitol City to Byzantium (at which point he renamed it after himself). He wanted the official religion of the Roman Empire to be Christianity and felt that Rome was too full of Idol Worship and the old religion and he wanted a fresh start/clean slate in already Christian Byzantium.
Alexandria? It was a very important city in the Roman Empire. It once held the world’s largest library (before the fire).
Antioch? It too had a secular importance in the Roman Empire.
(Similarly, in the U.S. certain cities stand out that people are familiar with: New York City, Los Angeles, San Fransisco)
Meanwhile, since those Original 5 Patriarches were established and recognized, other Churches in important secular cities have also be elevated to “Patriachates”. The Orthodox Church has continued to grow. “Newer” (using that term relatively, since some of these have been around for more than 1,000 years) Patriarches have been established: like the Church in Moscow, as well as small countries that are so small, rather than naming the Patriarchate after a city, they named it after the entire country: the Church of Serbia, the Church of Romania, the Church of Bulgaria and the Church of Georgia - all are valid Patriarchates.
Other places have Churches which are independant of Patriarches, sometimes called: Autocephalous…they include the Church of Cyprus, the Church of Greece, the Church of Albania, the Church of Poland, the Church of Czech - Slovakia, the Church of Finland and the Church of Estonia. The reasons for their independence various from founded by an Apostle to due to secular politics/oppression independence was needed.