I feel a lot of the people disagreeing here are making assumptions about your beliefs, missing the point, and then simply refuting you to refute you without providing explaination. I think this is a fair and interesting premise. I disagree with it and will ecplain why, though do note I am not invested enough to specifically look anything up so if I say something inaccurate, please evaluate if the logic falls apart or not.
I think the first part of your main justifications has been hard to refute. Most, if not all societies we have known have had religion or spirituality. However, I think your following conclusion, "those societies must have then used morality based on those religions", is where the flaw is. I think most societies had religion as a form of a "God of the gaps" and used it to explain phenomena they couldn't. I would say that is the main reason they did have it. However, that doesn't yet mean they didn't use it for morality. To see that, I'd ask you to look at Greek and Roman mythology, or as known to them, religion. Now I believe, Zeus turning into a swan and doing Zeus things doesn't have a moral (or not a useful one, it's mainly that Zeus is an asshole).. Likewise, Aphrodite turning Arachne into a spider didn't really inform some Greek moral of don't be too pretty, just showed Aphrodite is, for lack of a better word, a fucking jealous bitch. Let's similarly look at Norse mythology. Loki makes Fenrir and tries to kill other gods and generally does shenanigans. There's not really a moral attached to that, he kinda just does shit cus he's a hit of a dick.
My main point here is that while these religions existed, they did so to explain phenomena or were then essentially fanfic extensions of the reasons/personifications of those phenomena, and often were not the basis for morality of a culture (but very well likely were themselves molded by a cultures morality in a reversal of causation). Because Greece, Roman, and Norse cultures were more secular, they could therefore have stories without morals that just had assholery abound. Because the time around the formation of the Christian church was more tyrannical (now I'm guessing), the bible had much more heavy handed morals (ten commandments, 7 deadly sins etc).
I hope that was a better argument for disagreement. And, I don't think your premise was as outlandish as so many others are making it out to be, despite my disagreement.