Sorry, historically you are incorrect.
What we call the Byzantine Empire was called by the people who lived through it and by the Greeks today as "The Roman Empire" -- they were a continuation of the Roman Empire and the citizens called themselves Romaoi -- Romans
Their language may have become Greek and their dress too, but they considered themselves right until the fall of Constantinople as Romans -- hence the Turks called their 12th century kingdom in the south of Anatolia as the Sultanate of the Rum -- "Rum" being the Turkic pronunciation of Rome
"Rome was a Byzantine vassal" -- that's illogical -- as I said, Byzantine is what we Westerners call it, to the Byzantinians they were the Romans just taking back the eternal city from the barbarian Germanics
What did happen was that the Roman Empire was divided into West and East and even after the West lost its Roman Imperators, it was considered part of the overall Roman Empire -- barbarian kings acknowledged the nominal overlordship of the Imperator/Caesar Augustus
Justinian nearly succeeded in getting this back together but in the 8th century there was just the exarchate of Ravenna and the area around Venice left
The Orthodox church was the head church during that time. -- again an oversimplification. The 'Orthodox Church' and the 'Catholic Church' officially before 1054 were the same.
The Copts were formerly the Alexandrian church, along with Byzantium, Antioch, Jerusalem, and Rome. They weren’t called Roman Catholics or Latins either, but obviously the different churches ‘evolved’ into those distinct sects once Rome fell. -- sorry, wrong again
The first break-away was political -- the Assyrian Church moved away thanks to the Persian king using the Nestorian split as a pretext to separate HIS Christians from the Christian Roman Empire (Theodosius II in 395 declared Christianity as state religion). The Coptic split was the precursor of the next and I suspect, language had a large role -- the ones who formed the Oriental Orthodox were Coptic or Aramaic or Syriac or Armenian speakers
The Orthodox-Latin split was also in no small part due to politics and language -- by the 10th century and especially in the 11th, the East and West just didn't understand each other -- few Westerners spoke Greek and next to no Easterners spoke Latin, leave alone the barbaric Vulgate latin or Germanic languages
The reformatting in the 16th century was different from these earlier splits -- at least those which came after Lutheranism and Anglicanism, both of which had a political element and both of which retained key elements of orthodoxy like the sacraments
No, the problem with the 16th century movement was it opening the door to old things like Arianism (Jehovah's Witnesses) or Gnosticism (Unitarianism), Montanism etc.
Yes, I am aware that Byzantium is the Roman Empire, even if I didn’t hint at that. However that is my point. It’s hard to say that the Latins were the dominant Catholic church since before Constantine when you have the Orthodox church sitting within the preeminent Eastern Roman Empire while Rome was both a piece of the Empire for part of, and more or less on it’s back for a good deal of the other part of the last half of the first millennium.
And in theory yes, Rome was simply part of the old Empire the Byzantines wanted back, but that was their perspective. In practice Rome was never fond of Byzantine rule and strove against it even when it could barely take care of itself. Besides, Byzantium hardly held Italy long enough to be of consequence before the Germanics took a good chunk of it right back. The reality is the whole time Rome was scheming with the Franks and other Germanics to sweep aside the Arians, take Italy, and form something out of the old Western Empire clear up to the point where Pope Leo found his man in Charlemagne.
That’s why I termed it a vassal state, as it constantly chafed against Eastern Roman rule, as did the Germanics and Slavs as a whole. This is evident by the fact that at the first opportunity, the Latin church jumped right in with the Frankish Holy Roman Empire, a New Rome essentially where it would be the head church inside a real empire that would both crush it’s rivals the Arians and rival Constantinople. They even tried to expand that concept to the Slavs with the newly Christianized Bulgarian Empire to put the Byzantines against a two-front threat.
In light of that, I guess at the end of the day I look at it this way - the Latins can assert and pound the table about their supremacy all day, but no one is listening. History just doesn’t jibe with their story. The only way I see that changing is by attrition as the Muslims seek to exterminate the Christian church in North Africa and the Middle East, but that is a marriage of necessity and convenience, not acquiescence in my book.