A couple of paragraphs of important caveats before we get going. Fielding stats are still a work in progress, and tend to be subject to wild fluctuations over the course of a season (especially for starting pitchers, who appear in only 1/5 of the game). The increased usage of defensive shifts has also had a significant impact on the reliability of these numbers, especially for infielders. As the WSJ wrote: "Did a player make a play because he has incredible range, or because prescient scouting had him stationed in that area?" [Also of interest in that article: teams appear to save more runs with shifts than defensive talent]
It's likely the arrival of StatCast will eventually help redress the balance, since information is (slowly) becoming available for public analysis, rather than just for teams, which identify in detail where a player was when the ball was hit, when he fielded it, and how he got there.