You're still saying the same thing, only using a more palatable example. If there is no other way to obtain that sandwich, and given that having a sandwich is presumably better than not having a sandwich, especially if you're hungry, then the fact that the only way for him to obtain that sandwich was through war means that the war is justified, because he must do it to retrieve his sandwich.

The point is exactly what you said: there is no compelling reason for a war over a sandwich. But thanks to this line, you don't need one, the war justifies itself.

Of course, Lan is using it for a specific example where the lengths to which the Shadow are willing to go to in order to capture Rand, Mat and Perrin demonstrates that it is necessary to keep them out of the Shadow's hands. If the Shadow wants something that badly, then obviously the Light needs to keep it out of their hands.

But the problem is that he phrases that sentence as a general rule, not a specific case. And once you extrapolate what exactly such a rule entails, it's core insanity is revealed.