There is a claim that "the church was unified under the pope until 1054". But there was a "schism" centuries earlier than that, which is a far larger and messier divide than the 1054 schism between the Roman and Orthodox churches. It makes a lie of the "unified under the pope" claims of today's Roman Catholic apologists.
In fact, if you consider that "the papacy" didn't even come fully "developed" until Leo I (461 AD) -- and that it's still "developing" now under "Pope Francis", you've got the "unified Roman Catholic and Eastern Orthodox church under the pope" from say 461 AD - 1054 AD.
The churches in Asia, "the churches of the east", which had no idea that there was a "Petrine ministry" for many centuries, had as much of a claim to "apostolic succession" as did any of the European churches, and which grew far larger than any of the churches in Europe, before being snuffed out by Islam -- not in the 6th or 7th centuries, but the 12th and 13th centuries, likely a response to (a) the Mongol invasions (which were favorable to Christianity), and (b) the Crusades. (Not saying there were no massacres prior to the Crusades. But the Crusades exacerbated a bad situation).