The fact of the matter is that no matter what they called themselves at the time, their claim to Rome was de jure only. Their control was so far eroded that for historical reasons, their direct control of Morea and even functional overlordship of Athens is a bit of a stretch. Other countries claimed to be, and de facto were, stronger successors to Rome than the Empire in Constantinople. I agree that yes, the Roman Empire was the Roman Empire that fell in 1453, and all others are pretenders, but for gameplay purposes, Byzantium is simply a better name, as it sets them up as "merely" another claimant to the title, next to the Ottomans (who incidentally also didn't call themselves that), Russia and the HRE.
In fact, when it comes to Byzantium > Roman Empire progression, I would actually like for them to be disallowed from forming the Roman Empire via the decision, much like the HRE is, and instead get a name-change - not tag-switch - to "Roman Empire" after completing both the "Recover Lombardia" and "Rome Reclaimed!" missions. This would reflect their resurgence into being Rome not simply in name, having restored their prestige and strength, and make the journey from Byzantium > Rome feel more like a continuation, rather than taking a sudden hard left into fantasy territory. Well, a harder left than Byzantium surviving at all already is.
I hope that explains why I, at least, am against renaming Byzantium to Roman Empire in 1444. Nothing sinister about it.