You're not arguing in good faith. The title "King of the Romans" was bestowed by the Pope onto the Franks, who, again, had taken land that used to part of the Roman Empire. The pope needed to curry favours from the Franks and gave them the legitimacy they needed to rule Europe hegemonically. That title was passed on to the HRE and it stuck around. The Holy Roman Emperor didn't clam to be Roman, only to be the true heir of the Roman Empire. I don't deny their claim, nor do I deny Russia's, or the Ottomans'. The Roman Empire as a political entity was so large and so ecumenical in nature, that it provided legitimacy to many hegemonical nations to rule large swathes of land in Europe and the Mediterranean. I will however die on the hill that in the Medieval era, the only people to call themselves Romans outside of the city of Rome was the Byzantines.
The Romans of the King of the Romans absolutely claimed to be Romans.
As for the popes, you curiously seems to forget they were (and remain) popes of the Roman Church.
The bad faith is claiming that Greek had extra privilege to call themselves Romans while other "Romans" are disqualified. But it is easy to understand this misconception from people who clearly have no familiarity with Ancient Roman history.
You give absolute pre-eminance to the state continuity. "Byzance" deserve to be called Roman Empire because it is the same "state". If I stop trolling and start to discuss seriously, I inclined to admit it has a little value. But just a little due to the mix of civil wars discontinuity, the fact this "state" was not a "whole" "state" but merely half of its administration (which were regularly cut short through civil wars and coup d'état anyway) and how nonsensical the very notion of "state" is for pre modern era.
The big problem is very quickly, the Romans themselves distinguished between their national identity (Gauls, African, Italian, Greeks, Syrians, Illyrians, etc) and their shared Roman identity. This is something you can find in 4th century and 5th century authors. But even earlier still. A classic document to study in university is a letter from Cicero to one of his friend where he explain his feelings over his shared Roman / Arpinum identities.
It is in fact a fundamental difference between the city state of Rome and Greek polis. The Romans were relatively open to grant their citizenship to notable foreigners first, and eventually most people living in their "empire". Turning the "Roman" part into a legal statue rather than a national identity.
The other problem with the state continuity argument is it fundamentally misunderstand what was the "Imperium Romanum" during the ancient era, essentially an inequal federation of city states, and it misunderstand how "emperors" fit in their when the Roman Republic was de-facto destroyed by Octavius. Hint : the quotes around "emperors" are not just for decoration.
There is no ignorance in my comment. I am merely pointing out the historical reality that is at odds with your assertion. Have a wonderful day... preferably with a book.
Your comment is grossly ignorant. First because the de-facto capital used by the imperial administration had long changed before Diocletian.
Famously, Gallenius based his civil administration and his military comitatus in Milan during the height of the "Third Century Crisis". Before that his administration was based in Treves while his father was campaigning east with his own administration.
Your focus on administrative capital is a flawed one anyway. The capital was where the or an imperial court went. Meanwhile Rome the location and the city state continued to hold pre-eminence long after Diocletian. Which is why Constantius II visit of the city was a big deal for exemple. Among many other events.
