Maybe the confusion is about the word "privilege"? It's often used to indicate a special opportunity, but I took it to have the old meaning of "private law": an (exception to a) rule or service that is only available to a certain group. Going to war definitely wasn't that sort of privilege. A Greek polis would bring as many people as it could, even if all they could do was throw stones. Service as a hoplite or as cavalry also wasn't, going by Van Wees' argument. It was a civic duty, something that you were supposed to do with pride, but not an exclusive right.