I'm going to try and give my opinion on all of this (again)... I do believe that there's some nuance lacking in this discussion.
(1) The Byzantine Empire was, undoubtedly, the continuation of the Roman Empire during the Middle Ages. There is no clear-cut event where you can say that the Roman Empire stopped being “Roman” and started being “Byzantine”. The term “Byzantine Empire” was never used during its lifetime and it is a purely historiographical invention, though this does not mean that the term is illegitimate - it's purely shorthand for “the Roman Empire during the Medieval Era”. Historiographical names are not unheard of, though - the Yuan state, after being driven out of China proper, still called itself the “Great Yuan”, though historiographically it is called the Northern Yuan.
(2) The main ethnic group of the Byzantine Empire, and the one its rulers (for the most part) were part of, called itself Ῥωμαῖοι (Rhōmaîoi, lit. “Romans”). Medieval Westerners called this ethnic group the Greeks, and not a more literal translation like Romans. Undoubtedly this is tied with the messy history with the claim of succession to Rome, but really this is also just an exonym. I don't think it's that different from the Germans being called German and not “people-ish”, despite that being the literal translation of Deutsch.
(3) Citizens of the Western Roman Empire did not cease considering themselves as “Roman” for a while, even after the demise of the WRE as an entity. However - apart from the people who lived in the city of Rome itself -, by and large (there are still some Romance ethnic groups that still call themselves “Romans” or “Latins”, like the Rumantsch in Switzerland), they stopped considering themselves as Roman after an indeterminate amount of time had passed - I would say that already by the year 800 they did not consider themselves as Romans - but as Hispanics, Gauls, etc.; though they still called their language Romance, and would continue to do so for a few centuries more.
(4) The “Barbarian Kingdoms” that occupied the territory of the WRE did not completely up-end the Roman way of life. Many of these Germanic groups had already been long in contact with Rome, and had at least been partially romanized. The Roman nobility integrated themselves into these kingdoms, and indeed the ruling non-Roman class also became more and more romanized as well. The Roman state apparatus did not disappear and was adapted and integrated into these kingdoms' administrative structures - really, the “barbarians“ adapted the Roman state apparatus that was left in their provinces wholesale. Hell, even after their independence from Rome, they still kept on promulgating new laws decreed by Byzantine emperors. All in all, the Roman administrative apparatus did not cease to exist after the occupation by the Germanic kingdoms, and indeed kept on functioning well after the conquests. I should also say that, nowadays, the term “Barbarian Kingdom” is falling out of favor in historiography.
(5) Arguing about who “the true successor” to the Roman Empire, in 2025, is kinda moot anyways. As a history student (and hopefully future historian), it's a lot more interesting to look at how historical people perceived their own world, than for us present-day people to argue about the things they argued a thousand years ago. I'm not interested in which state or nation is the true and correct successor of Rome - I'm interested in how people in the Byzantine Empire, in the Holy Roman Empire, etc. saw themselves. Also, I absolutely loathe the “not Holy, not Roman, not an Empire” quote about the HRE - it is endlessly paraded by edgy teenagers who get all their history knowledge from r/HistoryMemes. It is an incredibly simplistic way of seeing history and it drives me up the wall - unironically, it's one of my biggest pet peeves.
My own take is this: the Byzantine Empire is the Roman Empire. It is wholly correct to refer to it as Rome or the Roman Empire, and I would prefer it if the game rule actually referred to it as the “Roman Empire”, instead of the “Eastern Roman Empire”, which is a bit of an absurd term in a context where there is no Western Roman Empire. However, the term “Byzantine Empire”, from a historiographical lense, is also correct.
I believe people take this too seriously, but that's not bad - I also take other minute things seriously. Though I would rather people not insult each other over a historical polity that's been dead for around 500 years.
On that note, I know that I got quite heated on some of my earlier posts in this thread, and I would like to apologize for that.