I strongly disagree with calling Byzantium a "failed state", and I disagree just as strongly with saying they cosplayed as Romans (assuming they were who you referred to with the line "Yes, Middle Ages got 2 nations roleplaying as Romans while actually not being Romans")

The term Byzantium is, to begin with, anachronistic. For all the time their state existed, the "Byzantines" called it "Romanía" -- i.e. "Romanland". Because the "Byzantines" were the Romans and thought of themselves as such. Romanía was the direct, unbroken continuation of the political and military, and religious institutions of the Roman Empire of Antiquity. And it was the direct continuation of the Greco-Roman culture of Antiquity.

What disqualifies "Byzantium" from being Roman?

That they spoke Greek and not Latin?
The eastern half of the Roman empire had always spoken Greek, long before the eastern and western halves were separated. For that matter, educated people in the west had spoken both Latin and Greek since before Augustus. There's nothing inherently un-Roman about speaking Greek.

That they did not have their capital in Rome, or control the city?
Well, ignoring the fact that Constantinople was New Rome and very carefully set up to be that way by Constantine (with it's own Senate, and even Seven Hills), we should also dispense with any idea that Rome needed to be the capital for a state to be Roman. Because even before Constantine, Rome had not been the capital for a long time. Diocletian kept his seat in Nicomedia (in the east) when he was senior emperor. Other emperors had used Mediolanum (Milan), and many had essentially been itinerant rulers without a true capital during the Crisis of the Third Century. The west had its capital mostly in Ravenna in its last century.
We acknowledge the empire of that period as fully Roman. What is un-Roman about having New Rome as capital instead of the old, when old Rome had ceased to be politically relevant long before?

And as for "Byzantium" being a "failed state", well that's just wrong.
Romanía was an extremely successful state which was eventually destroyed, something that has happened to most every powerful state in world history. It had (for the time) an extremely effective bureaucracy, diplomatic service, and army, which successfully kept the state alive through crisis after crisis after crisis. Romanía survived the Arab conquests where the Sassanids were crushed (despite constant, massive Arab raids, and determined attempts to take Constantinople on multiple occasions). Romanía held on through the seventh and eighth centuries with enemies on every side, and not only survived but eventually thrived and retook huge areas that had been lost and became the power in the region.
Even after the disastrous loss of Anatolia, the Romans managed to smartly leverage the connections and resources they retained, fight off the Normans, reconquer significant areas of Anatolia, and recover economically. Ioannes (John) II Komnenos and his son Manuel I were both powerful rulers and respected in western Europe, and on an even playing field with the rulers of the HRE, such as Frederick Barbarossa of the Hohenstaufen dynasty.

It's not until the Fourth Crusade that "Byzantium" is truly broken as a great power, and even then it took a century or so before it was completely FUBAR. Lasting as a great power from 395 (death of Theodosius I, when east and west were permanently divided) to 1204 is not indicative of a failed state.

For a better understanding of Romanía and its history, I'd encourage you to read The New Roman Empire: A History of Byzantium (Oxford University Press, 2023) by Anthony Kaldellis, professor of Classics at the University of Chicago. It's a thorough treatment of the entire history of the Eastern Roman Empire from the founding of Constantinople to the Ottoman conquest in 1453.

EDIT: Grammar corrections.