When I refer to a truly moneyless society, I mean a way of life which does not weigh the worth of a human being in money or property, but treasures that person simpy because he/she is a human being and as such, a member of, potentially, the most evolved species in the known universe.
Okay, you don't mean a moneyless society, you mean a society without society. (I'd also note that B doesn't necessarily follow A there. Maybe we treasure people not by money or property but by fertility or humor.)
Society (the way you use the term) is ABOUT setting up little tribal affiliations and packs of "us and them" and valuing people by some external (unrelated) criteria. What would you propose society would DO if it didn't give us the framework in which to evaluate people?
I'm assuming that by "society" you mean neither "economy","collection of rich people", or "political system" but "shared social moral system between a group of individuals" - which I find to be basically equivalent to "cultural mores" - which in order to function at all require that you BE A PART of said culture. Otherwise -- well you don't really share the morals do you?
Like most forms of utopias, and perfect societies, etc it has a hidden cost - which is that everyone has to get on board at the same time and then STAY on board. Then there has to be a way to enforce the conformity which is necessary for that kind singularity of purpose. Normally that kind of group solidarity is achieved through rituals, community events, and other shared practices. It's called communitas and you gotta get it if you want everybody to feel like they're part of some "bigger" than themselves.
Even if you could somehow structure your perfect society so that everyone is included and you don't have to have an "other" to pit all your member against - what will you do when somebody gets "off the reservation?" Gonna kill 'em outright? Brain surgery? Drugs?
Utopias are always the most dangerous societies to live in.
Eventually though you're just valuing people by how much they conform to your main principle there (of valuing everybody equally)- which of course is a violation of your main principle.