Well, it might be argued that Western Catholic clergy developed to be the bunch of power-grabbing, theocratic bastards we all know and love

also because it never met effective checks to its political encroaching till late in its cultural evolution, the time of absolute monarchies. Had the Popes lost the fight for the investiture, or the Roman Western Empire never fallen, the picture could have much more different, and Christian clergy developed a tradition of loyalty to the state instead of theocratic power-grabbing. See the Byzantine Empire and the Orthodox clergy, it shows that with different politcal conditions in Western Europe, Catholicism might have been rather different (I've always thought Europe might have been way better with Gregory VII and Innocence III being put on the stake by outraged Emperors).
Besides, the Roman Empire in the 2nd Century had the buds of an organized empire-wide secular bureaucratic service, had the Empire been more longeve it might have blossomed into something quite similar to the Chinese civil service, and that could have stabilized the Empire in the long term.
I think this kind of TL is quite feasible IF the Empire manages to conquer and Romanize all of Germania in the 1st-2nd Centuries (they never lose at Teutoburg and keep the momentum for the conquest of Germania). They bring the border of the Empire to the Vistula-Carpathian Mountains-Dneister line. Occupation of Germania spurs the Romans to develop the improved ploughshares to till the Northern soils effectively. Germania is Romanized to be as prosperous as Gallia. With the extra resources they gain from the new provinces, and they can spare from the way shorter border, they conquer Scotland, Ireland, Nubia/Axum, Mesopotamia and Persia, and bring the Eastern border to the Indus. At this point, further expansion to the Dvina-Dnieper border becomes feasible. By assmiliating almost all of Europe, North Africa, and the Middle East in her fold, Rome has expanded to her natural borders (The Atlantic, European Russia, the Sahara, the Indus, Ethiopian basin) and increased her economic/demographic pool (and drained the one of the barbarians) so much that barbarian invasions stop being a serious threat to her long-term political unity (apart from the occasional Hun/Mongol breakout from nomad Central Asia).
Such an Empire is still exposed to the threat of recurrent political disintegration from internal unrest, but if the Empire uses the breathing space from expansion to develop an effective universalist bureaucratic elite to check the military power, its long-term political evolution may shift from permanent political feudal fragmentation to a Chinese cycle of recurrent breakups and reunification or permanent divison into Western and Eastern halves. Checking how much the universalist legacy of the Empire was strong in OTL EUropean culture, if the Empire Romanizes all of Europe, North Africa, and the Middle East, it stands to reason it would become too strong in the elites to allow permanent disunity of the continent.
This will have massive consequences. It will butterfly away the Dark Ages to one of the relatively short periods of dynastic collapse and political disorder, and the expansion of the Arabs. Christianity might or might not still become the dominant religion of the Empire (if it does, it will be something like Orthodoxy) and be upstaged by Mithraism or another mystery religion. The Renaissance and the Age of Exploration most likely are anticipated by half or two-thirds of a millennia, to the 800-1000s.
I highly doubt the Diamond theory is correct to the extent that geopolitical factors effectively prevent the permanent political unity of the preindustrial political unity of Europe and Mediterranean basin to anything substantially larger than OTL national kingdoms. It might be well argued that Rome fell because it had not reached her natural borders, since thanks to her very efficient assimilation capability, the more it expanded up to them, the stronger it became. At the very most, those factors might have prompted the permanent breakup into a Western Carolingian half and an Eastern Byuzantine half. Still quite a different pathway from a dozen warring kingdoms or feudal anarchy.