Just typing those words reminds me of how abysmally ignorant I am about the East and its tangled glory. It was in so many ways a bulwark for Faith in Christ Our Lord.
Byzantium’s survival was undone by political and bureaucratic misrule. Perhaps the worst mistake was that of Justinian in sending Belisarius to defeat the barbarians and try to hold Rome — a gross waste of a military genius. Trade, diplomacy, and cultural diffusion would have worked in the long term to civilize the barbarian kingdoms and likely make them reliable allies against Islam.
It still is. The Holy Orthodox Christian Church lives on, carrying on the ancient tradition from apostolic times up to today.
As for ignorance, don't feel so band. It seems like western society doesn't want us to know about it. Or else western society thinks so highly of itself, it doesn't consider the 1,000 years of Byzantium worth looking at. Or maybe it's because it's impossible to look at Byzantium without staring CHRIST in the face.
My recollection of high school history is that you learn about ancient (pagan) Greece as a mere setup for Rome. Then when Rome falls, you get a brief mention of the split, but then rather than follow the trail of the eastern empire, it's immediately forgotten, and we skip ahead to Charlemagne, and follow the Roman/Germanic narrative, to the exclusion of the rest of the civilized world.
Meanwhile, Byzantine (Orthodox Christian) culture moved north into slavic lands -- Bulgaria, Ukraine, Russia, Romania, Serbia. The west, if you haven't noticed, seems quite prejudiced against those lands and peoples. They get treated like the white trash of the world by the western nations. But that's OK. I'm sure it will all work out in the end. Forgive me for running on.
The beginning of the Roman Empire was the annexation of Ostia, which (thanks to the Gauls burning the city and its written records) wound up being a prehistoric event, but it took place about 396 BC; Rome was sacked in 410 AD, but the last Emperor was deposed in 476 AD, a period of 872 years. This doesn't account for the various poorly documented territorial gains during the preceding generations, both before the last Etruscan king was booted around 509 BC and the period of slightly more than a century until Ostia's annexation.
Calling the realm of Constantinople an empire after the loss of Anatolia to the muzzie onslaught, and the piratical raids by the muzzies thereafter, isn't realistic imho. After the death of Boris II early in the 11th c, there was a more or less steady erosion of control over the eastern territories, mismanagement, tax increases, and the reemergence of what we might recognize as the Deep State.
Small successor states tottered on for a time (a decade?) even after the fall of Constantinople to the Turks, but they'd already gained territory in the 14th c via seagoing invasion of parts of the Adriatic coast of Balkan territory.
On a side note, I've often guffawed at the term "Byzantine" to refer to the eastern Roman Empire.