Nope. The Idea of Charlemagne becoming Emperor didn't originate from the Roman ideal but rather from a religious concept of that time. In Charlemagne, others of his time saw a great Leader as King David or Emperor Constantine were. As a result of this, he may have been the "renovated emperor of rome" as his title was, but his idea for his new empire was completly diffrent: All christian peoples should unite to become _the_ single and only people of christ. Thus his empire should become the empire of the christians, the "imperium christianum".
That continued through Otto I. who wanted to revive Charlemagnes Empire, thus the imperium christianum.
The first actual time, the head of the HRE saw himself as a "roman" was with Otto III. He was also the one to pursue the agenda of "renovatio imperii romanorum", so the renovation of the roman empire.
But i agree with you, surely the HRE should be named something like "Roman Empire" for 1000+ as it was called then, but then again, HRE is just the popular name for it (historians don't use it that often or point out that it is indeed an anachronism) and thats what ck2 players know it as.
Well historians actually didn't know what the ideas behind the Renovatio of Charlemagne was, you cannot just say what were the ideology of Charlemagne because historians debate this topic for centuries and no one know, because we don't have so many sources, apart from the official documentations where he is called indeed Roman Emperor in a mixed formula between the late roman and the byzantine ones, or the coins where he is represented in purple toga with laureles and with the inscription "Karolus Imperator Augustus".
Imagine that there are still people who believe that he really didn't want to be crowned Roman Emperor according to the myth of "Charlemagne came to Rome and he was not aware of the coronation". And that imho is a myth biased by a form of frankish proto-nationalism and then resumed by french nationalism later on, but that's only my personal interpretation of that passage of Eginhard (and that just to say that we didn't know for certain).
By the way... In the eyes of western Europe Charlemagne was the direct successor of Constantine VI, because in Constantinople there was a heretic emperor who was also a woman, and according to the Roman law and traditions (and the germanic ones), a woman cannot lead a juridical office like that of the Imperium, while the pope, a Roman magistrate in all effect (remember that at the time the pope was still appointed by the Senate of Rome, ie: an assembly of the Roman Nobles) had the legal ability to bestow a title like that, so he was in all effect Roman Emperor according to the ideas of the men of the time.
Further proof of that is that in the painting of Piero Della Francesca "The battle of Heraclius and Khosrow", Heraclius is regarded as a Roman and not "Greek" emperor, and so in the picture he carry with him the banner of the HRE.
Also I recommend the book "byzantine civilization" of Cyril Mango, you can read there that in those times the same epithets were conferred to the Estern Roman Emperors, like "emperor of all the world" or "emperor of the christians", the Roman Empire was "The Kingdom" and the emperors where supposedly the holy successors of king David, Salomon and Augustus at the same time.
Those conception of regal power was just "translated" from Constantinople to Rome again in the eyes of the westerners, and religious power and Roman ideal where inseparable in the medieval vision of the world.
