The Encyclopedia Brittanica does not seem to agree with you. In discussing the Empire’s position during the entire period of the Lombards it says:
‘Byzantine Italy was nominally a single unit, but it too in reality fell into several separate pieces. Its political centre was Ravenna, which was ruled by a military leader appointed from Constantinople and called exarch from about 590. Exarchs were changed quite frequently, probably because military figures far from the centre of the empire who developed a local following might revolt (as happened in 619 and 651) or else turn themselves into autonomous rulers. But the impermanence of the exarchs made it easier for their local subordinates to gain some measure of autonomy. The duke of Naples, the largest city of the south, was effectively independent by the 8th century, as was the duke of the newly formed lagoon city of Venice. The most important of these local rulers, however, was the pope, the bishop of Rome, for Rome remained the largest city of Italy and its bishop, in theory the spiritual head of the whole of Latin Christendom, had considerable status. Rome had dukes too, but they did not have the local support the popes had, and they remain shadowy figures. The popes, on the other hand, had a political position that in practice equaled that of the exarchs and lasted a great deal longer.’
This period started nearly 200 years before the Popes, with Frankish military support, subjected all the northern Lombards to Frankish or Papal domination. This is despite the fact that the Lombards had finally given up Arian Christianity, the factor that, together with their Barbarian heritage, had alienated them from the descendants of the classical western Romans.