No we didn’t, and I don’t think that it is even arguable that the Bishop of Rome was not the head of the western church, nor that the Bishop of Constantinople was the head of that church. Each was the patriarch of his respective church, and these were two of the five that formed the Pentarchy.
I can’t even tell which two you mean.
The Bishops of Rome and Constantinople were definitely having a dispute. Each was the patriarch of his respective church. That the messenger was a cardinal is irrelevant.
roughly a thousand years and counting
You’re mixing notions of “jurisdiction” here.
Communion between the churches was through their patriarchs; there is no “choose” by the members. The amount of “jurisdiction” of each patriarch in his respective patriarchate varied, and was certainly not the top-down monarchy of today’s west in any of them.
Not so much “required” as “consequence”: the head of a church, whether patriarch or other, establishes communion with the other heads. This creates (or breaks) communion for each entire church.
The schism opened by Rome with Constantinople didn’t immediately create schism between either and the other eastern churches.
AMDG
hawk