The importance of Jerusalem to the Moslems, as the site of the Prophet's Night Journey and the place to which he initially had the worshippers turn in prayer, would seem to indicate that the early Caliphs leaving the Eastern Roman Empire alone indefinitely was unlikely ; and if they demanded access to Jerusalem for pilgrimages to the Mount of Zion/ Temple platform as a venerated site this was unlikely to be granted. In the 630s Heraclius was busy trying to sort out the mutual antagonism of the orthodox 'Chalcedonian' Christians (ie those who followed the fairly rigid Catholic doctrine laid down at the Church Council of Chalcedon in 451) and the locally powerful 'Monophysite' communities in the Levant, and later created his own 'Third Way' theology which was supposed to be acceptable to both sides as a compromise , initially based on Chalcedon and later on a revised theology seemingly closer to the Monophysite version by re-focusing the wording of doctrine away from issues where the 2 sides were far apart in favour of common ground. But - as might have been expected - this only infuriated and was rejected by both sides and the use of state power to put the new version's supporters in senior offices of the local Church and to intimidate resisters backfired. The end of the 620s wars was thus followed by religious antagonism across the Levant, inviting for any new invader.
Please forgive me for truncating your post for brevity's sake; most of it I agree with and so I wanted to focus in on some key points. I'd agree that long term, conflict between the ERE and the Arabs is inevitable but
when would probably be key here, for both states. Given a few decades to recover, while the Arabs have become overstretched taking Iran, could result in a continuation of the Pre-628 status quo, in that the Caliphate simply assumes the position the Zoroastrian Sassanids formally held in the never ending stalemate of the Near East. The point you use to argue against it, of religious conflict possibly undermining the ERE, is the main area I disagree with your analysis.
Specifically, I would advance the claim that Monothelitism
did work as a compromise. Three of the five seats of the Pentarchy (Constantinople, Alexandria and Antioch) all got behind it, while Rome under Pope Honorius was, at worse, neutral. The main opposition within the ERE itself was from Palestine and this was later resolved, again with the approval of the aforementioned three, in the 630s; at that point, opposition switched to Rome as Honorius had been replaced but the Papacy was too dependent on the Emperor for this to be seriously challenging in the long run. There's also evidence that religious tension did not really exist as an aide to Islamic expansion, but was instead a post-facto justification/rationalization. I'll get to that next.
This factor of a religious confrontation arguably helped the Moslems by leading to some persecution and causing the Monophysite (and ethnically non-Greek)communities in places like Egypt to fail to help defend the Empire when the Moslems arrived in 638-41. This factor of religious and possibly ethnic / cultural opposition to the Empire's ruthless centralist policy in the 630s as a factor in its quick collapse in the face of Moslem invasion by smallish armies has been downplayed by recent historians. But it may have amounted to another incentive for the most adventurous and 'gambling' Moslem generals, eg the Caliph Omar and his lieutenant Khaled, to consider the risks in invading Palestine and then Syria and Egypt worth taking. To have the 'high command' in Mecca and Medina ignore the possibilities of success in attacking the Empire indefinitely you would need to alter Roman religious policy, and probably have a much shorter and less exhausting series of regional wars in the 602-29 period - and have a Caliphate keen to wage war elsewhere, eg extending their Iranian empire East and NE into the Indus valley and Central Asia in the 660s onwards, which would risk giving too much autonomous power and armies to local generals.
Robert Hoyland's
In God's Path, specifically argues against this interpretation in Chapter Three:
Modern scholars often argue, on the basis of later witnesses, that the Egyptians who were anti-Chalcedonian (that is, they rejected the creed agreed upon at the council of Chalcedon in ad 451) welcomed the Arabs and only the Chalcedonian Egyptians opposed them.8 However, John of Nikiu never once intimates that he or his fellow anti-Chalcedonians were in any way well disposed toward the conquerors. He also makes clear that the Arabs themselves were indiscriminate in their slaughter and that the disunity among the Egyptians lay not in sectarian differences but in how to face this challenge: whether it was better to submit and make peace or to stand and fight. “A great strife had broken out between the inhabitants of Lower Egypt, and these were divided into two parties.” Of these, one sided with Theodore, the commander-in-chief of the army in Egypt, who was determined to resist, whereas the other side felt that their interests were best served by negotiation and accommodation with the invaders.
And this indecision seemed to grip the very highest echelons of government. The elder son of Heraclius promised to send Theodore a large force in the autumn of 641 with which to repel the enemy. However, upon his premature death, his younger brother chose not to respect this promise, and furthermore he reappointed Cyrus, who had been Chalcedonian patriarch of Alexandria during the 630s, but who had been sacked for his conciliatory stance toward the Arabs. Indeed, it was known that Cyrus not only favored paying tribute to the invaders in exchange for peace but had recommended offering one of the emperor’s daughters in marriage to ‘Amr, “with a view to his being consequently baptised in the holy bath and becoming a Christian, for ‘Amr and his army had great confidence in Cyrus and regarded him with great affection.”9 This proposal had angered Heraclius’s elder son, but his successor gave Cyrus power and authority to make peace with the Arabs, to check any further resistance against them, and to establish a system of administration appropriate for the new circumstances
I feel at this point a general point should also be made that, what we know as traditional/doctrinaire Islam, might not have existed this early. Key components such as the Five Pillars, the Hadiths and the focus on Mecca came later; Hoyland puts the date on what we know as Islam as emerging about ~750 A.D. onwards. Non-Muslims, including Christian Arabs, Zoroastrian Persians, etc made up a lot of the "Islamic" forces of this time, and local institutions had to be co-opted into the Arab structure to make the conquests stick; Shoemaker goes as far as to argue early Islam was more of a "community of believers" than a defined religion distinct on its own, which explains a lot of the interesting qualities of it Jackson already touched on. Point being, I don't think it's sound to emphasize an overly "Islamic" guiding force to the conquests.