Claims of Byzantine caesaropapism are a bit overdone. As is usually the case, the history is pretty complex. For the Russians, at least from the time of Peter the Great, yes, the charge fits fairly well. Perhaps before, with the reforms that produced the "Old Believers."
But in Byzantium, no--if anything, the emperors ran into ferocious resistance from the Eastern Church, especially after 1204. Moreover, there were emperors like John II (1118-1143) espousing a "two-swords" approach to civil and ecclesiastical power.
More to the point, the papacy hasn't always escaped the meddling hand of Caesar, either.
One thing I will agree with is that Kemalist nationalism in Turkey has been an absolute disaster for the Orthodox Church. From the ethnic cleansing that pushed over 1 million Greeks out of Anatolia in 1922 (note especially the rape of Smyrna) to the little-known 1955 pogrom that effectively destroyed the remaining Greek population in Constantinople to the legal strictures like the most recent court ruling that have throttled the Ecumenical Patriarchate, Kemalist secularism has seen the steady dissolution of Orthodoxy in one of its former heartlands.
Like it or not, the Ottoman Sultanate was a far better deal for Christians in Turkey proper than the blood-and-iron nationalism of Kemal Ataturk.