That's true, except a trip from New York to California commonly occurs for around $400 or less. Does the EP hop on a plane for $400 or less?
Yeah, and at times like these I've found tickets to Athens cheaper than the NY-LA flights.
If I recall correctly, the EP has only visited the US twice in the existence of the church in America. Further, an American Patriarch would eliminate the fact that:
Well, you don't recall correctly. I've been present for at least two visits, and there have been others in addition. The current EP has, I believe, visited America at least 4 times.
1) The EP is under "house arrest" in a primarily non-christian country
Not true, but I know what you're getting at: the records, icons, relics, books, and other "things" of the Patriarchate are unable to be taken out of Turkey because they are considered cultural heritage items. Otherwise, he's free to move about as he wishes, although security does indeed follow him - but we are talking about Istambul, where there is an enormously high crime rate: yes, he gets targeted for hate crimes, but even if he wasn't there are many people who are not safe in that city.
2) Is not permitted to travel in public wearing any clerical identification
The Ecumenical Patriarch is one of 3 or 4 resident clerics in the nation of Turkey who is indeed permitted to wear his clerical garb in public. Visiting clerics theoretically can wear their own garb as they wish (I've seen an Athonite monk wear his full garb around the city), but usually visiting clergy of the EP do not.
3) Is required to be a Turkish citizen (eliminating a large number of possible candidates)
This wasn't much of an issue until the 1950's cleansing of the Christian population from Istambul and the subsequent shutting down of Halki Theological School. Before then there was a large Christian population in Istambul.
4) Would be able to minister to more than the 2000 people he currently has in Turkey.
He does minister to more than the 2000 people.
Why does this all matter very little? Because a bishop is to be concerned with his diocese. The only person who should be ministering to the flock in Pittsburgh is the Bishop of Pittsburgh; the president of the Synod that Pittsburgh sits on has very little to do with the decisions affecting the people of that diocese. Situations where a Synodal president exerts a wide influence over other diocese (such as 20th century examples in the U.S.) are exceptions to the rule.