As an Eastern Orthodox, I've never heard this line of reasoning.
Instead, what is usually discussed is the difference between primacy and supremacy. In the Orthodox view, among all the bishops, there were five who were regarded as preeminent based on the importance of their sees. These were:
The Orthodox recognized Rome as "first among equals", that is, as the preeminent bishop among the five great sees, and among all bishops everywhere. In disputes, Orthodox would from time to time appeal to Rome.
However, the Orthodox reject the idea that the bishop of Rome is on a level above or superior to other bishops. All bishops are the successors of the apostles. There's no reason to suppose that the bishop of Rome has an office that is different in kind from the office of bishop held by all the other bishops.
The significance of Rome is not the Petrine founding of the church there, but simply the fact that it was the capital city of the Roman Empire. In 381, Constantinople received the second place of honor after Rome, not due to any apostolic heritage, but because it was the second most important city after Rome.