The powers are real and limited, and the argument in Orthodoxy is about where the limit falls. The Patriarchate convenes pan-Orthodox meetings, coordinates common positions, receives appeals from other jurisdictions in certain cases, and has historically granted autocephaly to new national churches, which is how most of the modern Orthodox churches came to be independent. It does not appoint their bishops, legislate for them, or govern them.
It also holds direct jurisdiction over a large territory outside Turkey: Mount Athos, the semi-autonomous Church of Crete, the Dodecanese, the New Lands dioceses of northern Greece administered by Athens, and much of the Greek diaspora in Europe, the Americas and Australasia. The diaspora claim rests on the Chalcedonian phrase about bishops among the barbarians, read as covering territory outside any existing patriarchate, and other Orthodox churches read it differently.
At home its position is difficult. The Christian population of Istanbul is now very small, the Halki theological school on Heybeliada has been closed since 1971, and Turkish law requires the patriarch to be a Turkish citizen, which constrains the pool of candidates. The grant of autocephaly to a church in Ukraine in 2019 led the Moscow Patriarchate to break communion with Constantinople. The atlas records the rupture and the competing accounts of it without adjudicating between them.