The "evidence" to which Prof. Cole refers comes from, in fact, suppositions about what must have happened as a way of explaining the success of Islamic arms. Depicting these provinces as ripe fruits waiting to fall into the lap of the Muslims, this view does not give the Muslims very much credit for their own conquests.
There is not actually much evidence of local collaboration with or even satisfaction about the Islamic conquests, and there is more that tells us that the invasions were viewed very negatively. Coptic chronicler John of Nikiu recorded the coming of Islam as a disaster for the empire, to which Copts and other non-Chalcedonians retained strong allegiance. They just didn't like that their confession wasn't in control of religious policy and believed that God was punishing the empire because of the government's Chalcedonianism. There is some irony that secular historians have been reproducing a charge of anti-imperial disloyalty against religious dissidents in Byzantium that matched some of the official government views of these dissident groups.
Prof. Cole said:
Towards the end, of course it was the Byzantine Empire, the Eastern Roman Empire that ruled Egypt, there's some evidence that the Egyptians didn't really fight to retain that government when the Arab Muslims came in because the Byzantines had attempted to impose Eastern Orthodoxy in Egypt and the Egyptians were Coptic and had their own [sic]. So even with the Romans towards the end, I think they were weakened by their social policies.
It is perfectly understandable that Prof. Cole would say this, since this was a common view until not all that long ago. If you are relying on Ostrogorsky's classic text, you will come away convinced that this interpretation is right, and it does sound rather compelling at first. If you are thinking about the intersection of political and religious loyalties from a post-Reformation perspective and assume that religious dissidents would mobilise (or fail to mobilise) politically because of their religious sympathies or disagreements with the authorities, you are going to misunderstand the late antique and early medieval worlds rather badly. Neither Egyptian nor Syrian Byzantine subjects were organised or mobilised in the seventh century, and they would not have had much, if any, tradition of being mobilised for military service. North Africa has even less supporting evidence for religious alienation, since Carthage fell some time after all religious controversies between Constantinople and the west had been settled, which has led to some very imaginative but rather far-fetched claims of some enduring legacy of Donatism.
Their "failure" to fight did not signal a lack of loyalty to the empire as such, but rather reveals that antique and medieval imperial polities did not cultivate the kind of conscious political attachment to a state that might very well be expected in later periods. Particularly in the absence of effective political leadership or organised military support, armed resistance by the population was extremely unlikely as a response to foreign invasion. Cities would yield to invading armies because they wished to avoid sack and massacre, and not because they secretly wished for a chimerical "liberation" from religious oppression. The "ease" of the Islamic invasions was facilitated by Byzantine political and military weakness following the Persian War and particularly by specific Byzantine defeats on the battlefield. There is an understandable desire to find some "deeper" causes for such a momentous change in the history of the Near East, but there are good arguments that this change can be best understood through old-fashioned institutional and military history.
Cross-posted at Eunomia