But isn't the Greek-Melkite Patriarch Patriarch of Alexandria, Antioch and Jerusalem?
This is a different can of worms. The Greek-Melkites consider themselves to be 3 different Patriarchates combined. Hence, they have separate cathedrals in Alexandria, Damascus in lieu of Antioch, and Jerusalem. However, the Vatican only considers Antioch their proper Patriarchal Seat. Even more confusing is when the Vatican made the entire Ottoman Empire be the Melkite's canonical territory in the middle 1800's which is why some of their titular sees include those which belongs to the Patriarchate of Constantinople.
Precedence is established by age of Patriarchate; then by tenure (date of Ordination/Consecration) within.
Age has nothing to do with it. There are Deacons younger than me who are senior to me; and vice-versa.
My bad, I mixed up age with tenure.
The organizers could have taken into account the presence of Ecumenical Patriarch Bartholomew hence the Greek prayer. The Melkite Patriarch was then chosen because the Melkites would be the closest to Greek amongst the Catholic Patriarchs.
Then again, we may be just overthinking it and the Catholic Patriarchs could have very well discussed it amongst themselves.