I would simply say that the distinction between the objection to specific wars versus general objection is an important point. However, the reality on the ground may well be a specific war that is just totally unacceptable. To say, well, I'm being asked to kill civilians but I won't object if the United States were invaded and I would still volunteer, to me, that is a coherent theory that makes perfect sense. It's the same thing as the general law of assault: you don't always have to object to assaulting people if you simply don't want to assault people who are not assaulting you.
With respect to how we deal with false cases, that's done simply through winnowing them out, and that's done all the time. It's done all the time only in connection with the question, do you object to all wars? It could also perfectly well be done on the question, do you have a conscientious objection to this specific war?