I see more and more internet historians claiming ancient peoples were nearly as rational, irreligious and pragmatic as modern Westerners. How accurate is this?

I can answer this with reference to the ancient Near East - hopefully other people can chime in for a broader perspective!

It really depends on how you define those three terms. From a very basic standpoint, there's no reason to assume that ancient peoples' capacity for rationality, religiosity, and pragmatism was any different from ours. (I am assuming here also that, because you mention religion, by 'ancient' you mean 'Bronze Age' or thereabouts.) Although there's evidence that the development of our brains was affected to some degree by technological innovation,^1 on the whole what sets us apart from other apes is our highly-developed prefrontal cortex.^2 This part of our brains - we think - is responsible for our more advanced cognitive abilities, which include language, introspection, processing complex social information, and goal-oriented and long-term planning. These are the kinds of abilities we need for rationality (introspection) and pragmatism (goal-oriented planning). Religiosity is much more complicated, mainly because it doesn't seem to have a specific part of the brain dedicated to it. Instead, we think it might arise from a number of cognitive abilities working together.^3 In other words: we have the capacity for these because of the fundamental structure of our brains, and we've probably had it for hundreds of thousands (if not millions) of years.

Now, a complicating factor is assessing the extent to which we have evidence for these kinds of cognition and to what extent in turn these compare to our modern ability. I'm going to cheat slightly and focus on the ancient Near East - partly because it's my field, but also because it's the oldest civilisation on the planet of which we have written evidence. Archaeology can show us (or suggest us) religious awareness (e.g. Göbekli Tepe), but texts let their authors speak to us directly. I'll go through each in turn.

Rationality is extremely well-attested in the Old Babylonian period. You might have seen that article about the Plimpton 322 tablet, which shows perhaps an advanced mathematical capacity superseding our own (or perhaps not quite...), but certainly proves that they were capable of computation and abstraction, and that their mathematical awareness was well-developed. Their astronomy, too, was solid: as early as the 17th century BCE we know of a major astronomical work, Enuma Anu Enlil, which describes in painstaking detail phases of the moon, annually-recurring astronomical phenomena, and planetary and astral movements. It is true that these observations were often tied in with omens and religious ideas; however, this should not be taken to mean that they were irrational. On the contrary: within the paradigm that existed in Mesopotamia, the existence of the gods was taken for granted, meaning causal links between the (non-empirical) divine world and empirical observations were logical for them. In other words, their rationality was perfectly coherent, and analogous to our own, within their scientific paradigm (which happened to include a bunch of gods).

Religiosity and irreligiosity are somewhat more complicated because of this fact, because the Mesopotamian philosophical paradigm naturally included the gods. To them, asking whether they believed in the gods would be like asking a modern Westerner whether they believe in gravity: we don't believe in gravity, it's just there, it just exists. To the Mesopotamians, then - and in fact probably the Greeks and Egyptians as well - the question of belief in a deity did not compute. Looking at it that way, Mesopotamians certainly weren't irreligious: but they also weren't necessarily religious as such, since theist/atheist weren't really categories in the same way as we think about them.

That is not to say, however, that the acceptance of higher powers and their cult were the same thing; and actually, we might be better off looking at the practical ways in which people engaged with religion. And then, yes, we find a lot of evidence of personal piety - take this hymn to Inanna, the goddess of love and war, whose virtues are extolled in no less than 274 lines of dedication. Take Esarhaddon, king of Assyria, whose reputation suffered because he required frequent oracles of encouragement from Ishtar, the goddess of his capital Nineveh; the fact that apparently his subordinates considered his wanting divine approval for his actions a bad thing here is interesting. Anyway, it wasn't just the elite - commoners, too, were actively involved in religious practice.

Although we thus have strong evidence of cultic practice, at the same time we know that not everyone took it as seriously. The strongest example I can think of is that of Esarhaddon's father Sennacherib, who absolutely despised the Babylonians; unfortunately for him Enuma Elish, the text that was recited every year at the new year's festival, heaped praise on Babylon's national god Marduk. As part of quelling yet another rebellion, he razed Babylon to the ground in 698 BCE, and instructed his scribes to rewrite Enuma Elish with his own national god Ashur taking the place of Marduk (see Kämmerer & Metzler 2012). This Assyrian version of Enuma Elish shows that cultic practice was at times dependent on political motivation - given that the Assyrian elite was happy to completely repurpose a Babylonian religious text to suit their own ideology - and thus gets closest to something we could call 'irreligiosity' within the boundaries of the Mesopotamian religious paradigm.

Finally, pragmatism. There are many ways of understanding this, but I'll give an example of Babylonian Realpolitik that I think approximates the concept quite well. The Assyrian Empire had always had a lot of trouble with Egypt. Although Esarhaddon - clearly despite his reputation - first succeeded in conquering and pacifying the region under Assyrian control, and his son Ashurbanipal retained control, the empire collapsed not even a century later and was succeeded by their mortal enemies, who formed the Neo-Babylonian Empire. Their most well-known king was Nebuchadnezzar II, who was quite worried about the resurgent Egyptians, who had quickly rebuilt their kingdom after the fall of the Assyrian Empire. The standard doctrine for the Babylonians, as it had been for centuries, would've been to meet them halfway for a battle, after which the winner would take control of that region and rebuild it, and so on until one of the two was conquered and incorporated. The Neo-Babylonian strategy, particularly under Nebuchadnezzar II, could be called pragmatic: because any army invading Babylonia would need supplies on the way (it's a long way to march all the way from Egypt), he decided to simply take away the supplies. As a result, much of the southern Levant was technically incorporated but practically invaded and laid to waste (famously, of course, Jerusalem), to deny the Egyptians the infrastructure to march on Babylon. (In the end, it was of course the Persians and Medes from the east, not the Egyptians, who ended Babylonian hegemony - so much for that plan.)

Hope this helps - please let me know if I can clarify anything!

Bibliography:

Kämmerer, T.R. and Metzler, K.A., 2012, Das Babylonische Weltschöpfungsepos Enuma elish. Münster: Ugarit-Verlag.

Everything else is linked directly.