1. WAR does use the "eye test" but against a standard. Many many fans advocate the Eye Test, but not one I have ever heard of has any sort of standard. In fact, too often the eye test degenerates into "I saw that guy make a couple errors once." Sometimes, it just evolves into the Reputation Test, where you hear a guy is an excellent defender so if he does nothing wrong when you watch him, hey, it must be true.
2. Using your instincts leaves you with quantifiable opinions, right? If I asked you who is the best defensive CF, you might say Bradley or Pillar or Buxton or Cain or some other candidate. What would you say if I asked you to support that opinion? Even i I asked you to support Benentendi/Bradley/Betts OF over Martinez/Benintendi/Betts OF, how would you support the opinion - probably shared by many including me - that it was better? This isn't an attack on you or anyone. It's the nature of what we see when watching games and how we all watch them given the massive imbalances in the players we see.
3. WAR most definitely has flaws, but it still is absolutely more encompassing than we are fans are able to achieve by watching games. Especially watching games on TV, where you don't have any option on what part of the play you watch. I see plenty of people talk about the jump an outfielder gets on flyballs - good or bad. That type of evaluation - important for defense - is something you rarely if ever can see from a televised game.
4. In the past, you have often said WAR has too many moving parts, and any system that has more moving parts is more likely to break down. Maybe true, But look around your house. I bet you have a car and not a horse. I bet you have a washing machine and not a washboard. I bet you have a refrigerator and not an icehouse. I bet you have an oven and not a rotatiing spit over a flame pit. I bet you have an air conditioner and not a hand-held fan. Every device I mentioned has more moving parts than predecessor I compared it to. Every device I mentioned is more likely to break down than its predecessor. Yet they all have another thing in common - they all work better than their predecessor, too.