It's the native Iranian dynasties that are the crucial factor here. I'll quote at length from the beautifully eloquent historian Peter Brown in his superb little 1971 book,
The World of Late Antiquity
"The Persians, therefore, had never developed that fierce sense of a religious identity that had kept the Ummayyads at arm's length among the Christians of the Mediterranean seaboard. Khusro I had taught the
dekkans, the courtier-gentlemen of Persia, to look to a strong ruler in Mesopotamia. Under the Arabs, the
dekkans promptly made themselves indispensable. They set about quietly storming the governing class of the Arab empire. By the middle of the eighth century they had emerged as the new backbone of the Islamic state. It was their empire again: and, now in perfect Arabic, they poured scorn on the refractory Berber who had dared to elevate the ways of the desert over the ordered majesty of the throne of the Khusros."
They don't make historians like THAT any more
Essentially, the reason no Syrian or Egyptian upper class survived into the Arab era is the fact that it had already largely been decapitated by Diocletian. The aristocracy had largely moved to Constantinople, and what remained, the lower gentry, had probably already been substantially damaged by the Persian invasion. By the time of the Arab conquests, the only real source of leadership for Christian communities in the old Roman Empire were the bishops, who, naturally, did not have an interest in taking over the government of the Caliphate.