I understand being frustrated with WAR. One felt more in control of a baseball conversation when you could look at HR's, see one guy had 33, and another had 21, and come to the conclusion that the 33 HR guy is "better" than the 21 HR guy. Easy peasy.
You can't just watch a game, look at the box score, and see who contributed what. But you're playing a little Skeptical Paradox game, here, pretending that nothing can really be known. WAR isn't computed by some guy drinking beer in front of his TeeVee. Defensive statistics are computed based on zone ratings that have nothing to do with the eye test. If a fielder is standing on spot X, which is measurable, and a ball is hit to spot Y, that is 15' from the fielder, with a hang time of 7 seconds, we know a lot about the catchability of that ball in play. We know how fast a runner can run from one base to another, how fast a player can throw a ball, all of this is quantifiable, and has nothing to do with the "eye-test".
Players on offense create runs. Players on defense prevent runs. Creating runs are pretty quantifiable. You hit the ball, and get to a base by the time the play's over. The same happens when measuring defense. Again, completely measurable and quantifiable from a fairly robust data set. When running the formula for Pythagorean wins, a win is worth (roughly) ten more runs scored than the opponent. I'm mixing models, here, but if Jose Abreu can create 5 more runs on offense than average, and can prevent 5 more runs on defense on average, that would be worth one win, or 1 WAR. Again, mixing models, but that's how players are evaluated against each other.