My query is fairly multi faceted. Was it viewed as a resurrection of the old Western Empire, or was the Western Empire viewed as having existed all along, and this was just the end of a very, very long interregnum? In terms of Rome as a source of political legitimacy, does this represent a continuation of the tradition seen in the reigns of Odoacer and Theodoric? Am I over thinking this and it was just a power grab without much consideration of 'Romanness'? What role, if any, has the coronation played in arguments about when exactly the peoples of western Europe considered the Western Empire to have ended?
I partly answered your question here, although admittedly that answer focuses mostly on England in the period. Nonetheless, the answer isn't radically different in that early medieval political aspirations hinged on Romanitas less as an immediate continuation of Rome, but as a valid successor through the continuation of the facets of Rome which had been worthy of admiration.
It was well known that Rome had failed and there was no way for Charlemagne to claim that he was simply the next Roman Emperor, but there was tremendous prestige in resurrecting Rome, reforging the empire and being the new Roman Emperor. It's worth mentioning, of course, that Charlemagne crowning himself as Emperor was not taken well by the Byzantines, who had been maintaining the Roman Empire perfectly well all this time without him, thank you very much.