What would a pod for avoiding Gregory's execution?
A simple POD is that the Mahmud decides to accede to the Sheyhulislam when he refuses to sanction the execution.
How do you think the Patriarch surviving would influence the Revolutions?
It would diminish support, I imagine - Gregory had excommunicated the rebels three times, while his survival would reduce anger felt by Christians throughout the OE.
Would a more Neo-Byzantine Revolution take hold as opposed to a Neo-Hellenic one?
Well, it'd obviously have to be a Neo-
Roman Revolution (Byzantine was principally a foreign term), but I think there's major problems. The principal problem with a revolution that espoused "Roman" identity IMHO was that the situation "on the ground" (e.g.
in Greece proper) was a lot different from the way it's been generally imagined. Keep in mind that there were several different factions in the revolt, including the educated class that promoted a Hellenic identity (these were people who were generally descendants of Byzantine nobles and held influence throughout Ottoman lands), a number of brigands and warlords in Greece proper who would have considered themselves
Romans, as well as a substantial Arvanite contribution. Keep in mind that the second and third groups had warred savagely in the past; Kolokotronis in his memoirs even recounted how his father had stacked pyramids of Arvanite heads. As such, a revolution that promoted a Hellenic identity could reconcile the latter two groups against all of the "Turks" of Rumelia.
Is the killing the Patriarch just too tempting and "gettable" a target for a Muslim-led government with pro-Muslim biases being stressed out by Christian rebels to avoid?
There was no precedent to executing a Patriarch for the rebellion of his subjects: the Ottomans didn't even
depose the Patriarch during the revolts of 1567-1572, 1611, or even the Orlov Revolt that had happened a couple decades earlier - hell, the Empire was even more "stressed out" by the rebels, given that they were simultaneously losing to Russia in a concurrent war. Furthermore, all precedent had established the independence of the
Rum millet, which is precisely why the Sheyhulislam had refused to sign the Sultan's
fatwa and ultimately paid for his life with it.
I fear people of the Ottoman side had a gut feeling this would release their frustrations and regardless of his personal conduct, Gregory was an attainable scapegoat to "show them we mean business."
Not really - it was more-or-less Mahmud II's own project, opposed by both reformist and conservative elements within the Ottoman government.
In OTL were the ulema calling for the execution from the beginning of the rebellion with the Sultan wavering and then eventually going for it?
No, they were not. They would have little reason to do so - after all, Gregory had excommunicated the rebels three times...