Okay, this is going to be a really long one. I'm just going to crush into the debate with my minimal historical credentials to say some stuff about the Byzantines (which I fervently care about, as you can probably tell). I'm going to preface this by saying I'm wholeheartedly rooting for them to be a separate civ. That being said:
There is no such thing as "Roman culture" outside of a colloquial sense. Roman culture stopped being the culture of Latium since the times of the republic since lands completely foreign to Latin Romans were gained and incorporated to the state. Rome had always been multicultural with no one singular focus or arbiter of cultural influence. Imagine having the US, but every state speaks a completely different language and has completely different customs and traditions. The whole "Greek culture influenced Roman culture" is in terms of high art and among the higher echelons of society (and even there with some friction from conservatives).
In that sense, the eastern half of the empire was always predominantly influenced more by Greek culture and most people spoke Greek as a second language instead of Latin. That had been the case ever since the time of the Diadochoi because of the emergence of Hellenistic culture in the area. "Byzantium" didn't speak Greek out of nowhere, they did so because the people living there had been doing it for over 500 years.
The defining reason why the Byzantines were the legit Rome and not simply calling themselves that is that there is an unbroken line of continuity between the ancient Roman empire and themselves, so the people who would ostensibly create a "new culture" for the empire had already been generating that culture for centuries before that. To give you an analogy: If the US was struck by a tsunami and the western half was swallowed completely leaving only the eastern to survive, would it still be called the US or not? It's a purely political issue and has barely anything to do with culture or language. If it did, then we wouldn't have Macedon separate from Greece or Australia separate from England (unless someone had decided to represent Australia with its actual natives).
Claimants to the Roman legacy such as the HRE, Russia or the Ottomans don't meet the necessary requirements. The HRE had an emperor appointed by the Pope which is a huge no-no in Roman politics. The head of the Christian faith was the Roman emperor, so appointing an emperor as a religious authority was a big issue, hence the Pope became a more political figure later whereas Orthodox patriarchs didn't. Russia's claim to the Roman legacy was some Palaiologina (female Palaiologos) marrying a Russian prince. The Russians had no Rome to inherit; not land-wise, not politics-wise. And of course the Ottomans weren't even Christian to inherit it. To be Roman had become synonymous with being Christian at that point. You can view the Roman emperor as a Christian "proto-Caliph" of sorts.
There is no sense in separating "Roman" from "Byzantine" emperors, they are one and the same thing. The fact people have a hard time deciding where Justinian belongs should be a clue, since he is part of the transitional link between the popular imagination of "Roman" and the popular imagination of "Byzantine". The answer is simple: Everyone before him was Roman and everyone after him was Roman as well. The division is arbitrary, even if for the purpose of the game it serves a good purpose gameplay-wise. If we had to have this division imposed, then Constantine would be purely Roman. Other than building Nova Roma, he didn't do anything else politically to actually define a Byzantine empire (one could argue no one ever did, but I'm not going to hark on that).
Rome was secular; Byzantium was deeply religious (one Medieval writer wrote that one could not buy bread in the marketplace of Constantinople without overhearing debates on Christology). Rome imitated Greek art; Byzantine art was innovative. Roman military was centered on infantry; Byzantine military was centered on cavalry (and navy). Rome spoke Latin; Byzantium spoke Greek.
Rome was never secular. The supreme religious authority of Rome in the pagan days was the Pontifex Maximus, a political office that you could pass as a career politician by simply applying for it. Julius Caesar was a Pontifex Maximus as well. Rome was also hardly secular in terms of state practices: During the second Punic war, the state imposed measures from the Sibylline books in order to regain favour from the gods, including burying a Greek slave alive in the forum of the city. Roman warring traditions, political decisions and all sorts of other processes also had to be "approved" by good omens and religious practices such as sacrifice and visiting temples of the appropriate gods.
"Byzantium" being deeply religious was as much of a continuation of the ancients as it could get. Even if you overlook Rome's religious practices and persecution of "hostile" religions, Greece has had a history of execution against people accused of blasphemy against the gods and their vibrant forum debates on ideas of gods/God, particularly neo-platonism from a point onward (which was also the philosophical foundation of all Christian theology).
Byzantine art was as innovative as Roman art was. The most defining characteristics of Byzantine art are domes, mosaics and religious icons. The first two had been features of Roman art and architecture since antiquity and religious icons were a continuation of the ancient pagan tradition of divine figure iconography. Small statues or illustrations of deities and even philosophers were a common household accessory. This is why icons are mostly a thing in the east since their initial iterations were very much reminiscent of worshipping them, unlike what Irene of Athens would have you believe.
Roman military was heavily reliant on cavalry as well in the late stages of the empire. Cataphracts are not a Byzantine thing, Romans took it from the Parthians and used it, hence Byzantines also inherited it. And their army became more centered around the infantry towards the later stages because the loss of much of Anatolia where the finest war horses came from. Palaiologian armies would have virtually no cavalry, Komnenian armies would have very small amounts and the only notable elite force would be the Archontopouloi, and that for a very short period of time. The elite force of the army was for a long period of time the Varangian guard, an infantry military unit.
And as I've said above, Rome didn't speak Latin, the western half did. The eastern half has always been predominantly Greek-speaking as a lingua franca. When Pontius Pilate supposedly spoke with Jesus, they would be speaking Greek. When an Egyptian merchant would bargain with a Levantine shop-owner, they would be speaking Greek. Knowledge of Greek was an essential requirement for serving a political office in the east, which is why Heraclius was Greek-speaking whereas the then imperial administration did not.
And technically Macedon was a kingdom that was located north of the peninsula where all of the Greek city-states were and was politically separated from Greece. Sure Greek culture influenced the nobility, but the rest of the people had nothing in common with their southern counterparts. As for Germany, The biggest territory of the Holy Roman Empire was the Kingdom of Germany and encompassed much of of modern day Germany at the time of Frederick's reign, so I don't have much of an issue with that.
Not related to the Byzantines, but just making some corrections here. There was no "Greece" in antiquity. "Hellas" was a place in Thessaly where the mythical ancestors of the Hellenes, "Hellen" came from and supposedly connected all Greek-speaking tribes (because that's what they were, frankly). There was no political institution uniting Greek-city states, there were shifting alliances throughout their history so everyone would be politically different from the other. Macedon was no less separated from the political happenings of the peninsula than Epirus was (which was also a similar kingdom). Not to mention city-states that were just as removed from the Greek power politics as Macedon was, namely most Greek city-states in Sicily and southern Italy.
The reason Macedon was particularly singled out is a) wartime propaganda because none of the other Greek states likes to lose and b) Macedon's government system which was not just being a kingdom, but an absolute monarchy. Absolute monarchies were deemed barbaric by Greeks because of a cultural shift and their contact with "barbarians" who tended to use that system. Sparta was also a kingdom just like Macedon was, but unlike Macedon or Epirus, the Spartan kings had limited power due to the noble and popular influences on the state. That was considered tolerated, but an autocrat like the Macedonian king was seen as backwards.
There are plenty of arguments why pretty much everything about Macedon was Greek even on the popular level (art, religion, given names etc), but the most telling is the Macedonians' own attempts at lifting that "barbarian" taint from their name. Macedonian Olympic athletes would constantly try to prove their "Greek-ness" in order to participate, among them king Alexander II (or was it the first? Can't quite remember). What can be indeed be said is that the pre-Hellenic substrate of Macedon was most likely not Greek, but that's not unique. Cyprus' natives before Greek colonization were not Greek, same for Crete, the Cyclades and so on. You can distinguish it thus by labeling it "Hellenized", but if it barks like a dog, wags its tail like a dog and eats its own **** like a dog, then wouldn't it be logical to call it a dog?
Lastly, a correction on the HRE comments: There was no kingdom of Germany, neither as part of the HRE nor ever. Germany in the HRE was a bunch of semi-independent duchies, principalities, kingdoms and archbishopic seats. Most of those were German-speaking, hence the association, but there was no real German identity let alone a political entity corresponding to that. Not that I disagree with Frederick being a German leader, but it does take a rather weird leap of logic along presumed sociolinguistic lines in order to present him as such. That goes even beyond the current Byzantine debate, in fact. It's sort of the equivalent of making Constantine XI Palaiologos the Leader of Greece. Yeah, he was Greek and the lands of the empire he ruled were only Greek at that point, but wouldn't it be odd to make that association in-game?
TL;DR: Yes, the Byzantines are Roman. Yes, they should be their own civ in the game. No, it's not a cultural thing.