Archaeologists in 4000 AD

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.

"middle period" is pretty optimistic

History began in 2022, so middle seems about right
- Umm akshually
  
  History is uh social conshtruct that only appearsh whenever (weshtern) hishtorians wantsh to
  I am shmart liek that...
"Late period" will be Mad Max style.

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.
- yeah no I'm out of my depth here intellectually, but it sounds interesting, mind elaborating?
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
- religion is essentially ideology and ritual

"And as you see, children, this is part of the worship of the God they call Capital"...

I use it to beat my partner in Tetris

domestic violence in tetris is never the answer
They must be a total blockhead

https://ianwrightsite.wordpress.com/2021/11/25/dark-eucharist-of-the-real-god/

The mention of Friedman brings us to another feature of the capitalism-as-religion hypothesis: the economic specialists who function in a way very similar to ‘theologians’. Not only do they debate the core doctrines of capitalism, and not only do they develop new ‘schools’ or ‘churches’, but they also advise governments in much the same way that priests, ministers and theologians used to advise European governments of the past. And where you have doctrines and theologians, you also have sacred places for worship. These places may vary – stock markets and banks come to mind – but one must show due reverence to them and engage in the appropriate acts of worship. Individual life too has its rituals of this new religion, focused on spending and saving, on buying and selling.

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.