Certainly the peasants also need to trade with each other, regardless of the presence of an army? Money facilitates that trade. You can barter chickens or goats or whatever but at some point nobody needs more of those either. Especially because chickens have to be fed.
When the thing being traded has only abstract value it becomes money, even if the tokens are chickens.
I think an argument could be made that armies are just a jobs program. In a way money necessities jobs because of the increased economic capabilities of that abstract trade token. With more efficient trade the peasants have less to do so they need ever more complex jobs. An army (and the military-industrial complex) provide those jobs.
I think Graeber's theory of the origin of money may not be universally accepted (not sure how accepted it is); but I think it's true that the 'common sense' theory isn't actually based on much historical evidence, but is just "an argument that can be made", as they say, which somehow became the "common sense". That's not a very reliable way to accurately describe history. All sorts of things that people in different times assumed "must be true because it just makes the most sense" turn out not to be. "seems simplistic" or "seems more plausible to me" is not a reliable guide for distinguishing truth from fiction, and what seems more plausible to me may not to you.
Money is a natural consequence of trade in general.
You are a ruler with an army. That army has to be fed. It can just violently take what it needs from the population. But that is disorderly and makes people mad at you.
So, instead, you issue money to the soldiers. By why would the population take this money in exchange for food? Even if it's made out of a precious metal, what does a peasant want with gold, when nobody else wants it either (unless maybe they want to make something out of gold, but who can afford that). Well, you say they need to pay you taxes in this money you have issued. Now they've got to get some of it, and the soldiers are the ones with the supply. So you give them food in exchange.
I haven't read that account of the origin of money anywhere else (and I'm not sure how much consensus there is around it among relevant ancient historians), but I have read that taxation originates generally from the need/want to maintain large/standing armies. Not just that you have an army for the purpose of exracting tribute by force; but that you had an army for some other reason (conquering "someone else's" land or defending "yours"), and now you need to extract resources from "your" population to maintain it. This idea is referenced a bit in OP in the context of Europe: "But from around the late fifteenth century, war became more expensive… "