As early as the First Council of Constantinople, the emperor tried to get the bishop of his city to replace Alexandria as the Eastern primate. This was opposed by Pope St. Damasus, who reiterated the traditional ordering. This was tried again at Chalcedon, but again, the Roman Pope (St. Leo the Great) vetoed it. However, when Alexandria rejected the definitions of Chalcedon and separated from the Church (leaving only Rome and Antioch), Constantinople filled the void and the Greek Patriarchate of Alexandria was instituted and was essentially a vassal of Constantinople. Jerusalem was also separated from the jurisdiction of Antioch at Chalcedon and fell more under the influence of Constantinople, especially when the emperor later made it a patriarchate. The rise of Islam also significantly weakened Antioch to the point where it became significantly reliant on Constantinople.
So by the time of the schism, Antioch and Alexandria were pretty insignificant compared to Constantinople. If we apply EO logic on this, the question should not be why Constantinople instead of Antioch, it should be why Constantinople instead of Moscow, since Moscow is now the most significant see among the EO Churches.
Of course, as Catholics we believe in a theological basis for Rome being chief, not just a political one, which is why Rome remained first in rank despite Constantinople being a more significant city. But for the EOs, there is no theological justification for Constantinople being first, so I don’t think there is any real reason for it to remain so when Moscow is de facto first in honor (see how everyone lined up in the recent schism between Constantinople and Moscow).