The Byzantines were as aligned with the "Orient" as Alexander was. They held onto territory in Italy until a full 600 years after the West fell (once Belisarius recaptured it in the 600s). They had active diplomacy with the Holy Roman Empire and other states in the West up until the 4th Crusade, basically.
You've completely glossed over the Renaissance of the empire around 1000 AD (550 years after the West fell) under John Tzimiskes, Basil II, etc. You can blame this on the decline of Abbasid power if you want, but even if the caliphate _technically_ held territory in Syria/Lebanon, there was no question as to military superiority. The Byzantines raided with impunity and got tribute from the local emirs.
Constantinople lost 90% of its population after the plague of Justinian, and a recurring series of outbreaks over the next centuries, plus a loss of manpower after the desperate end of the Sassanian wars under Heraclius (which Heraclius technically won, but both empires were so diminished that they were easy prey for the new caliphate). It rebounded. Never to its maximum, but it was back near 750000 under the Macedonians.
So, the empire was ok until 660, lost territory, rebounded in the 10th century, and was arguably at its best organizationally and culturally until the real decline started during Manzikert.
It lasted 1000 years. Please read something other than Gibbon, or at least listen to The History of Byzantium (which follows on where The History of Rome left off). History is not a static field, and the Byzantines are rapidly becoming rehabilitated as more documents come up.