To respond to what you said, Get_off_my_lawn:
To points 1 and 2, which are interesting and worth considering: I don’t know if and how priests convinced themselves that they weren’t lying. It only matters to me that they saw for themselves that what they were saying was false. If I’m chanting, “May you eat well Enlil” (that’s from the Sumerian song ‘Honored One, Wild Ox’) and then I go into the sanctuary and statue-Enlil doesn’t eat his offerings, I am confronted by that disconfirmation. I can shut it out, you’re right, I can do mental gymnastics to get around it, but… I still know that I said something untrue, even briefly or momentarily. I have similar claims from Egypt and China (“The spirits are drunk,” “Food of all kinds hast thou [statue-Amun-Ra] tasted”) that imply falsehoods priests knew to be falsehoods from observing temple ceremonies.
I don’t know, and I don’t try to speculate, about what priests said to other priests when they saw for themselves that statue-gods or spirit-gods did not consume their offerings. I’m willing to admit that they could have read into statements they saw on temple walls (Horus: “I have received your offering… I have drunk the wine”) that gods dined non-demonstrably, which scholars sometimes argue. In Eastern Zhou China, and I admit and discuss this for a few pages, elites sometimes said gods dined on the essence of the offerings (that’s found in The Guanzi, The Zuo Commentary, etc.). Essence is invisible, sometimes spirits are said to be invisible, therefore it is possible priests during the Eastern Zhou believed their own claims. Other claims about spirits, however, from The Mozi and The Book of Documents among others claim spirits appear visibly and audibly in temples (“A wizard carrying a cane appeared and said…”, “On the day of the sacrifice to Wu Ding, there appeared a crowing pheasant…”). Priests cannot have believed that sincerely.
Not every priest was lying about gods, and I want to be clear about that. In many Egyptian and Mesopotamian texts, often the divine feasting act (the god dining) is not described. The author instead passes over it without comment. Such persons are tacitly admitting that gods did not dine in temples, but they aren’t lying about it.
There is still a contradiction in many Mesopotamian, Egyptian, and Chinese sources, and that is the fact that statue-gods or spirit-gods did not dine demonstrably, as they are sometimes said or implied to. I am not the only scholar to notice this contradiction as I’ll call it:
Marc Linssen (2004): "The ‘feeding of the gods’ remained the most important purpose of the divine meal in Mesopotamia. Although the foods and drinks offered were re-distributed after the offering ceremony to priests, temple personnel, and others, an official cult maintained up to the end of civilization in Mesopotamia the fiction [italics mine] that the divine meal was consumed by the gods, or rather their statues, and that offerings served no other purpose (129)
Dale Launderville (2003): "So even though the cultic system arranged for the food and drink offered in the cult to be consumed by humans, the religious outlook maintained that deities consumed these provisions” (140).
To point 3: I don’t think sources are too sparse for an argument to be made. The sources I use are as old as the practices they describe, and were written by contemporaries; I read over 125 (many come from The Electronic Text Corpus of Sumerian Literature, https://etcsl.orinst.ox.ac.uk/). They survive on clay tablets, oracle bones, papyrus scrolls, temple walls, etc. which were found or remain in situ. They provide compelling evidence, at the very least, that what priests and kings said happened to the offerings did not happen to the offerings.
To point 4: you know that’s my argument, I won’t belabor the point. I would only add that besides power, elites wanted the food and drink offerings they received from slaves, peasants, and other citizens (the wives of elites, military commanders, etc.) and they ate and drank them after the ceremonies.