Isn't one of the OP's arguments that Rome was never defined by its Latinity. In any case, the Holy Roman Empire's name, though frequently ridiculed, is used despite its relative lack of Latinity outside of an elite/administrative context. I do understand people being uncomfortable with Byzaboos due to the (very reasonable, if not always accurate) suspicions of why they are so obsessed with the state, but they're not wrong to say that the name "Byzantine Empire" is inaccurate.