In principal I agree, but Costantinople had a sounder claim.
In 298 the emperor Diocleziano split the empire in two different administrative parts, the western one, culturally latin, and the eastern one, culturally greek.
The emperor Costantine I moved the capital from Rome to Costantinople in the 320s, the city was called also New Rome and the greek inhabitants of the eastern empire called themselves "Romei", and the balkans have been called then Rumelia a term that spread even further under the Ottomans. Even today many turks use the term "Rumeli" for balkan people (as Balkans is a term introduced two centuries ago by a german geographer).
By the end of the 4th century the split was basically complete and the latest Latin-born emperor of Byzanthium was Giustiano in the 7th century.
I think that the history of the Eastern Roman Empire has enough "de jure" claims for the inheritance.
This also connects to Russia.
By the fall of Costantinople in 1453 no country claimed to inherit the throne of the eastern roman empire but Russia. The Tsar Ivan III married the last granddaughter of the last emperor (Constantine III) thus uniting the bloodlines. The crown of the byzanthine empire was then given to the Tsars as it was the last country in Europe of greek-orthodox descendance (visible still today, both in religion, culture and language) thus claiming to inherit the throne of the roman emperors more than a thousand years after.
I do absolutely agree with you about the legitimacy, I don't think there's necessarily any nor I think it's relevant, but I find this extremely interesting nonetheless.