Jaime's contention is that if religion had never been invented, all of history's atrocities would have happened anyway (or perhaps, different but roughly equivalent atrocities, committed for other "reasons") because the cause of the atrocities is "human nature." Joe's contention is that religion enables at least some atrocities that would not be enabled without it, i.e., if there was no religion, would the Aztecs really have went ahead, invested lots of work and resources into building pyramids so they could drag thousands of captives up the steps and sacrifice them to...an idea that never occurred to them?
One problem with this debate: a world in which "religion had never been invented" would be one in which "human nature" was itself different. Religion is itself a product of human nature: authoritarian tendencies, over-active agency detection, a tendency to see the external world through social goggles and relate to it on that basis, and so on. For example, "Thor" is much easier for us to understand and relate to as a concept than the equations governing the behavior of electrostatic energy. This despite the fact that the latter are actually much simpler in computational terms, because you don't have to code a human-level AI to model the behavior of electricity. And to create an accurate computer model of "Thor" as god of lightning, you'd still have to accurately model the equations anyway, since they're derived from how atmospheric energy actually behaves, regardless of whether or not there's a "Thor" involved. In other words, "Thor's" human-level intelligence, emotions, etc. are entirely superfluous for modeling the behavior of atmospheric electricity, but it's still easier for us to conceive of "Thordidit" as an explanation for lightning than the physics, because of the way we naturally see reality through person-colored glasses.
So, if we try to imagine a world where religion had never existed, the humans there would have to be different creatures than the ones here. Their brains would work differently, so even if we could send a trans-dimensional probe into a parallel universe where religion never existed, and see that it was a better world, Joe could say, "See, I'm right," and Jaime could say, "No, I'm right because their 'human nature' is different than ours."
Instead of using "religion had never been invented" as the comparison, let us instead imagine a world in which religion had existed, but humanity had shaken off its shackles at some point in the past (i.e., prior to the atrocities we're discussing). Say, if Pharaoh Akhenaten had singlehandedly discovered naturalism and trounced the Egyptian priesthoods decisively in debate, and used his position to spread atheism throughout the world of his day. His writings spread from Egypt, toppling religious beliefs to the point that they became a tiny fringe in a secular world, opposite to the way that atheism was a tiny fringe in a religious world, as happened here. So, for the following centuries of human history there would have been a secular culture equipped with historical memory of religion, how it worked, and intellectual antidotes for its causes.
Would this have produced a better world, one with fewer atrocities due to the greater difficulty of generating public sanction for them in the absence of concepts like religious faith, gods who should be obeyed without question, the Divine Right of Kings, an afterlife, and so on? Even though I think there still would have been atrocities in such a world (e.g., Stalin and Mao were both able to perpetrate democide without religion, though it might be argued that both were able to cash in on the "benefits" of populations conditioned to unquestioning obedience by religion), I think there would have been fewer atrocities, and certain kinds (e.g., Aztec human sacrifices, people sacrificing their firstborn children to gods) wouldn't have happened at all.
I think the case for this is all but self-evident. If religion could not spur the commission of atrocities that would not have happened without it, and such things would "happen anyway" due to human nature, despots wouldn't use religion so much. Religion is expensive. Pyramids and cathedrals aren't cheap, nor are the gilded priest-castes who use them. If Jaime's view is correct, medieval kings could have launched a brutal conquest of the Holy Land by saying to their followers, "Let's do this for money and power!" instead of needing a Pope (Innocent III) to proclaim, "God wills it!" The Aztec kings would have motivated their armies and killed equal numbers of captives without having any need to share power and wealth with a priest-caste who convinced people that the sacrifices were needed so that the gods would keep the Cosmos running. And so on.
Which means: religion would be entirely superfluous and useless as an institution for bending people's wills to the service of tyrannical regimes. So why would said tyrannical regimes have invested so much wealth, human labor, etc. into religion in the first place? Why would the kings tolerate the existence of priesthoods whose wealth and power rivaled their own, when the priesthoods served no purpose the kings could not accomplish on their own with wholly secular regimes?
I think the conspicuous wealth, power, and resource-consumption (temples, sacrificial offerings, etc.) of religious establishments all over the world and throughout all recorded history is abundant proof of religion's vast power to brainwash people and compel obedience on a mass scale. If religion did not have such power as religion, nearly always outstripping non-religious attempts to do the same, religious establishments would not be nearly as ubiquitous as they are and have been historically. If religion had no effects over and above "human nature," how could it compete with secular alternatives that could achieve the same ends (atrocities, utility in propping up tyrannical regimes) without expensive temples, sacrificial offerings (animals, grains, captives who could otherwise be used as slaves, children, virgins, etc.) and a wealthy but superfluous priest-caste? If we look at history's great atrocities, the ability to brainwash people and compel obedience on a mass scale are always crucial to their commission. Is not the ubiquitous success of religion itself proof that religion is extremely useful (more so than secular ideologies) to accomplish those ends?