The imperial revival of the IXth century have multiple causes and was variously perceived within Carolingia itself.
(For clarity's sake, Roman refers to the city of Rome; Eastern Roman or Imperial to the Eastern Roman Empire)

Until the early VIIIth century, Roman papacy enjoyed a steady, if rocky, relationship with the Roman Empire as well as it became the religious center, if symbolical, of the western kingdoms. But the inability of Constantinople to protect what remained of central Italy and especially the Roman Duchy (that roughly corresponded to Latium) whom popes began to gain responsibility of; and the imperial support of Iconoclasm led to a significant break away of Rome, both religiously and politically : coinage of the duchy ceased to mention the emperor, the papacy marking thus its independence.

The Lombard pressure in Italy only got more important however, and while the conversion of the kingdom to a Nicean creed allowed the popes to negotiate or even to have the king giving them back some cities, the general feeling was that Lombards would eventually take control of the Exarchate and of the Duchy. Finding a new protector was part of the Roman diplomacy : Peppinids (otherwise known as Carolingians after Charlemagne's reign) managed to unify the Frankish realm under their authority since decades, but also tied a strong relationship with the Church, while benefiting from the immense prestige (at least at the eyes of Rome) having not only defeated Arabo-Andalusian raids twice but having carried war against them.

Thus, Rome supported the forfeiture of the Merovingian dynasty and the consecration of Peppin III and his family in 751. As soon as 752, the pope Stephanus called the Frankish king for help, and eventually for the first time in history, the pope crossed the Alps to take refuge in Frankish Gaul. At Quierzy, Peppin makes a symbolical gesture of alliegence, promising to restore the pope into his states (expended to include what remained of the Exarchate, that recently was taken by Lombards), which was done in 754, sealing the alliance of the Papacy and the Frankish king, who received the title of "Patrice of Romans".

Charlemagne didn't departed from this policy : Frankish kingship since the VIth century was heavily characterized by its Biblical and Christian identity, kings being "new Davids" or Solomons; and as his father before, Charlemagne stressed the importance of reforming Frankish church into something more canonical (literally so, as councils set the adoption of Roman liturgy in Gaul), calling or attracting clergymen from all Europe (Aquitaine, Spain, Britain mostly) at his court.
Not that the Carolingian religious policy was systematically on line with Roman pontiff's, which is particularly the case for the Council of Frankfurt where the Frankish church and Charlemagne held a uncompromising stance against the Second Council of Nicea when the Pope did the contrary, but overall Charlemagne had the support of the Latin churches and clergymen and was obviously a necessary interlocutor on religious matters.

Eventually, Charlemagne was everything Papacy needed in an emperor : strong enough to protect it, acknowledging a direct and special relationship with the pontiff (being the , abiding by Roman liturgy and conceptions seen as normative of an universal Christianity, etc.,
What seems to have carried the final decision was the involvement of Charlemagne in what was a Roman affair, the pope Leo III having been ousted of the city in 799, Charlemagne giving him shelter and forcing his return on the population.

Before we go to the matter of the coronation, you might have noticed something : we didn't mentioned the relation between the imperial title and the Roman Empire, neither what Charlemagne or Frankish aristocracy's perspective.

The reputation of Romans among Franks was contrasted : there was a difference made between ancient and medieval Romans, the latter being often considered as "Greeks", but Greeks that had the government of the Roman Empire since it passed to them. The popularity of the Empire significantly declined in the Latin west since the VIIth century, mostly for religious reasons but as well due to the decline of imperial gift diplomacy towards western courts. Ancient Romans were seen in a much more favourable way, the "people of Romulus" being credited with political, cultural and material achievements, but they were also seen as an oppressive people defeated by Barbarians who replaced them, especially Franks (people still considered themselves as Romans in Aquitaine, Provence and central Italy, but these were defeated peripheral areas which the Frankish aristocracy intermarried or even replaced).

On the other hand, the alliance between Carolingians and Papacy was articulated around an important principle : the Carolingian king, as much as were Merovingians, was the chief of the Frankish church or rather he was the temporal chief of the Frankish bishops as a "new David", at the imitation of what existed in the Late Empire or Eastern Roman Empire. The religious split of the VIIth century reinforced this power, now seen as coming from the Roman primacy but as well the Frankish royal power itself.
Carolingian kings and Charlemagne having imposed their hegemony not only in a divided Frankish realm (both Frankish sub-kingdoms and peripheral principalities) but also outside either by conquest (Saxony, Pyrenean March, Lombardy), pressure (southern Italy, Slavic petty-states) or even sheer influence (although this is true mostly for clergymen in Britain, Ireland, Spain, etc. and much less so for the lay politics if at all), their religious power was no more limited to their kingdom or the conquered Lombard kingship.

What Charlemagne had to benefit by taking the imperial title from a Greek empress (who, for Franks, couldn't really pretend to hold the function both because of her gender but as well because of the "infamous deposition" of her son) was to legitimize his universal rule over Christians, a divinely ordained imperium that befallen to the Kings of Israel, the Romans (the people of Romulus, Augustus and Constantine), to Greeks emperors and eventually to Frankish kings.
Alcuinus himself wrote that the royal dignity of Franks was greater and more bright than the imperial but as well the pontifical dignity, the king being the chief head of the Church and the only one able and legitimized to defend her.

The pope wasn't ready to give up his primacy and his main role, only to become the main chaplain of Charlemagne : while Charlemagne's reluctance to be consecrated emperor as recorded by Eginhard is probably a literary topos of Latin imperial literature (the agency of Carolingian palace highlights a quasi-imperial display), Leo III didn't followed the same ritual than what was practiced at Constantinople but instead used one which stressed a bit more the pontifical power and importance over the making of an emperor : the acclamation-coronation-proscynesis (prostration) order was reversed to give the main role to the pope.
It's unclear, however, if it's all that happened, as Roman sources are scarce (the Book of Popes is strangely silent on it) and Frankish sources mention other elements such as the presence of delegate from the Patriarch of Jerusalem.
At the least, we can say that Romans and Franks had a different conception and idea of what it meant to restore the imperial title in the West, Pope and King meeting an uneasy common ground for their different (and conflicting) interests doing so.