The assertion that Charlemagne had no idea he was going to be crowned is taken directly from the Vita Karoli (Charlemagne’s biography) written by Einhard. However, this text was written at least 17 years after the fact. It shouldn’t be taken at face value because it probably was a literary attempt to highlight the emperor’s virtuous humility. That this event had great propagandic value also becomes clear from a few other sources. The Royal Frankish Annals from around 807 depict Charles as a saviour riding into Rome to save the pope from his enemies (Leo III was in the process of being deposed by his enemies in Rome before Charlemagne stepped up for him in 799), while Leo’s biographer (writing after the pope’s death in 816) downplays his problems and places the pope in the driving seat. It is also important to remember that 8th century Rome was a religious and not a political centre. It did not have the power over the catholic church and the ambition to control secular rulers like it developed from the 11th century onwards. The coronation endowed Charlemagne not with more power, ‘only’ with more prestige, which drew on the powerful religious and classical–imperial resonances of the city but remained ill defined. For Leo II, it mainly meant shared prestige and through that protection from his enemies at home. He did not gain control over Charlemagne or anything like that because he was the one crowing the emperor. This also a claim that only arose during the Investiture Conflict of the 11th century.

The vagueness of this situation does however not mean that Charlemagne wasn’t planning anything. A few sources from the years directly preceding the coronation refer to dealings Charlemagne had with the Byzantine. A manuscript from Cologne probably written in 798 refers to an occasion ‘when messengers came from Greece to hand over the empire to Charles’. The weight given by the Franks to the position of Byzantium is confirmed by the Annals of Lorsch, which justified the coronation with reference to the fact that the Byzantines were ruled by a woman (the dowager empress Irene), and argued that the position of emperor was therefore implicitly vacant. This is almost certainly the earliest of our narrative sources, and probably represents a version of the official line developed by the court upon Charlemagne’s return from Italy.

Another interpretation is that Charlemagne’s imperial status was devised as a way of making his rule acceptable to the Italians, to whom the imperial title had the most resonance in post-Roman western Europe. Since the 6th century campaigns of Belisarius, southern and central Italy (including Rome itself) had been accustomed to an imperial presence either directly or mediated through the Byzantine exarchates. During the 8th century however, Byzantine power was retreating/collapsing in the region and this may have created something of a crisis of authority for Italian elites. Charlemagne’s bid to acquire imperial status may have been designed to answer that crisis and enhance his own status in and influence over the entire peninsula, while he only really controlled the northern part, by appealing to Italian models of authority. In this context it is significant that a Sicilian embassy visited Aachen in 799, that an official from the island is known to have defected to Charlemagne in 800 and that, according to the Byzantine chronicler Theophanes, Constantinople’s greatest worry about the Frankish king’s trip to Rome in 800 was that he intended to take over Sicily and the south. Nothing came of it however, as after 800 the Carolingian empire did not expand anymore as Charlemagne and his successors were mostly preoccupied with strengthening their rule over the lands they already controlled.

I mostly adapted this answer from the awesome book The Carolingian World (2016) by Marios Costambeys, Matthew Innes and Simon MacLean. It is by far the most authoritative and in-dept overview of the Carolingian empire available to day.