That wasn't because of the loss of Egypt, it was because the empire was relentlessly hammered by an enemy with 10 times its power and revenue. Nobody starved, it just used Crimean grain instead - and had already made changes in its supplies before Islam because Egypt had been lost to Persia.
I'm not arguing that Egypt wasn't a serious loss, but I don't think the empire can hold it, at least not with the Caliphate in charge of everything in between Anatolia and Egypt. Most Egyptians considered Constantinople enemy #1, not the Arabs. Independent it had a chance. The Arabs took it with a teeny little army - that wouldn't have been possible if the Egyptians had resisted in any way, and the Arabs were at quite a loss as to what to do themselves since it hadn't occurred to them that they could just walk in and take over.
The empire changed into it's "cousin" because a) it had been hit by a 2 year famine followed by b) the Black Death with killed a third to half of the population followed by c) a 20 year war with Persia and the Avars followed by d) the emergence of Islam and the resultant 200-year body slam that entailed. The empire was forced to reorganize into the Thematic military state in order to survive. There should be a pre-a) that it had been wracked by centuries of heresies and their incredibly violent suppression.
After losing Egypt to Persia and after all the doctrinal disputes, Constantinople was essentially a hostile occupying regime. Under those circumstances Egypt was a strategic liability, not an asset. Over time maybe it could have become an integral part of the empire again, but not after the emergence of Islam.
Well it seems we don't entirely disagree, but as I said if Constantinople can manage to keep Egypt out of Arab hands its not just going to give it up to the monophystites, heresy or no heresy.
Well that's the thing; they didn't. They simply collapsed and the empire suffered an economic catastrophe of a crisis that lasted on-and-off for the next century and a half. The polity which emerged was recognizable as a distant cousin of the old, but not much more.
Between the loss of Egypt and the plagues the ERE's population simply...disappeared.