You need to understand that money became much more than a tool facilitate trade. Nowadays it's become a way of consolidating power and controlling resources, which amplifies inequality. When you say:
There’s no rule that says an anarchist society needs to be moneyless.
That's not exactly the full truth. Getting rid of money right this instant would be a logistical nightmare. But there are already so many things we don't need money for it's hysterical. Being an anarchist today means you question authority constantly to either deem it legitimate or not. In todays age money is a great tool for you to hoard resources and otherwise wealth. So the question for us is not "How fast can we get rid of money" but rather "What kind of system can we implement in order to make money practically useless."
Edit: In other words, lets say you have 100 Dollars, you decide "I wanna buy some bread." You stumble upon a commune that eliminated money entirely and traded based on need. You see two people are actively trading bread and fruits... with no profit incentive whatsoever. Just kind of trading because "ehh, I have enough apples for myself, here you go." Wont that interaction make you completely question those 100 bucks that you planned to use?
No and no. Since I took food as an example I'll use that because we all need food and it's the most important product that needs to be addressed. The way we produce today industrially is 1) centralized, 2) profit-driven, and 3) heavily dependent on exploitative labor and fossil fuels. In contrast, the model I’m proposing would decentralize food production, it would be very emphatic on local autonomy, and be organized around mutual aid and shared responsibility rather than trade or barter.
Instead of massive monoculture farms and supply chains thousands of miles long, food would be grown close to where people live - through networks of community gardens, small-scale permaculture farms, and cooperative distribution. The tools and materials needed, (yes, even some that are industrially produced) could be made in worker-run, federated workshops where production is democratically planned and prioritized based on need, not market demand.
So no, it’s not barter. It’s not trade. The example that I tried to give was not fully "you give me X, I give you Y." (I know, I did a shit job of explaining it) It’s a gift-based, need-based economy rooted in reciprocity (what we already see in disaster response and indigenous food sovereignty projects). It’s about building systems where everyone has access to what they need without having to earn it or bargain for it.