The thing is that you are taking a split of 1054 and projecting it backwards
in 315 AD when Christianity was no longer outlawed (but not the state religion), there was no large Church at Constantinople -- the 4 Churches were Rome, Antioch, Alexandria and Jerusalem. Jerusalem was nowhere near the force it was due to the destruction in 69 AD -- it sunk into nominal significance. Ditto for Antioch -- difficult to defend and threatened with sacking by the Persians -- it was sacked twice by the Persians in 250-256 AD
There was basically Alexandria and Rome. The Alexandrians did produce the larger chunk of Church Fathers, it was a center of Christianity and if you look at real numbers you would think that they would get primacy. Yet, the bishop of Rome was given primacy -- in the sense of first among equals and in many councils as the deciding vote
And this way it remained during the first decades of a non-persecuted Christianity in the Roman Empire
now, by the time Christianity was made state religion -- in 395 AD, yes, Byzantine had waxed in importance and so too had the Patriarch of Byzantine
Yet, there is still no separation of orthodoxy -- Alexandria, Antioch, Jerusalem, Rome and Byzantine are united
Theodosius II elevates Byzantine to the pentarchy, purely as the Emperor was sitting there -- and yes, that made sense
From that point on there is a tussle between Rome and Byzantine -- but not a tussle between Latins and Orthodox as you are projecting backwards
The Byzantine Church has some problems like Patriarch Nestorius etc.
And, in all of these cases, the role of the Patriarch of the West remains as the primus inter pares -- there is no doubt of that
To a specific sense that you make of "a dominant Church within the Catholic Church" -- if you mean one ruling over the others like the medieval papacy, then yes, you are correct. If you mean one given spiritual 'first among equals' place, then no, you are wrong
You forget about the first split over the veneration of icons, coming out of the second council of Nicaea. After that the Latins and Orthodox pretty much de facto go their separate ways, capped by Pope Leo crowning Charlemagne as Imperator Romanus.
And no, at the point, at least on the surface, they weren’t hostile, but the division was there, and had been there since the Lombards. It just widened over time due to various events, but in reality Rome had always sought it’s own destiny over the other churches. In many respects the Latins did what the Protestants did centuries later vis a vis Constantinople.
Which is fine in my book. They were the proper church at the time to win and organize the Germanic tribes in Western Europe out of paganism, even if they had their missteps and corrupt Popes. As far as the current, I have great respect for Popes Benedict and John Paul II. They do and did tell it like it is.
I only have a problem with those Latinophiles that seem to want to rewrite history to make Rome the dominant church since Christ, when clearly they were not for centuries, in principle or actuality.