The priestly class would congregate in tall temple-towers where they would divine the will of the market by performing arcane rituals involving cultic items called "spreadsheets" that were considered very powerful but whose meaning has been lost to time.
The prophecies divined from the spreadsheets would be used to determine the food rations of commoners.
Is there a case in history where society was heavily influenced by what was later considered a religion, but wasn't at the time and didn't claim to be?
You could maybe make an argument that the way religious people differentiate themselves from pagan tribal beliefs or cults fit this categorization. The way Catholics and Orthodox Christians treat their saints while calling pagan beliefs barbaric is quite humorous.
Well considering the concept of "religion" is a thoroughly Western idea and has no equivalent in most other ancient cultures (see "The Heathen in his Blindness...": Asia, the West and the Dynamic of Religion by S. N. Balagangadhara) I would say almost all of them. "Hinduism," to just cite one example, is a constructed "religion" that did not understand itself as thus and arguably didn't even exist until Westerners constructed it.
Would religious cultures even consider themselves "religious" since it was the de facto ideology of their society?
You'd have heretics from other cultures, but I don't know if religion would even be a definable thing unless separated from the structure of your society.
I am speculating here and thinking about the concept, I do not know either.
Would religious cultures even consider themselves "religious" since it was the de facto ideology of their society?
Going by the catholic church example they seemed to, like most religions, arrive at a point of "We say this is how it is because you gotta have faith" or similar which to me would be the distinguishing feature to this, where it's "actually it's all scientific" (ceterus paribus, i.e. we making shit up
The After the End mod should have this as feature of the Coffee Current with Columbia, sort of akin to the trade in incense from Arabia to the Mediterranean. Make it applicable to the Capitalist religions that follow the Profit.