You're missing the whole point of BABIP when you say "BABIP is one of my least favorite of the new sabremetric stats because it depends largely on the pitcher."
The whole point of BABIP is the following (from Baseball Prospectus' glossary):
Batting Average on balls put into play. A pitcher's average on batted balls ending a plate appearance, excluding home runs. Based on the research of Voros McCracken and others, BABIP is mostly a function of a pitcher's defense and luck, rather than persistent skill. Thus, pitchers with abnormally high or low BABIPs are good bets to see their performances regress to the mean. A typical BABIP is about .290.
Further research has shown that a very small minority of pitchers have a small influence on their BABIP, maybe .280 instead of .290, but a pitcher with a BABIP of .213 (which is where Rich Hill was 3 weeks ago) is experiencing a lot of luck.
Now, as you point out, I have not seen every out made this year with Rich Hill on the mound. But the defense doesn't have to make "26-27 diving plays in the field" to influence BABIP. Think of it more like when Hill's on the mound and the hitter rips a hard grounder right to the shortstop for a groundout. Maybe the next time the hitter hits a grounder just as hard, but it's 10 feet over and goes for a base hit. Hill got the ground ball both times, but he can't MAKE the hitter hit it right at the fielder every time.
Hill's BABIP at different levels over the last couple years:
2004 Dayton: .296
2005 Peoria: .313
2005 AA: .297
2005 AAA: .300
2005 MLB: .306
2006 AAA: .282
2006 MLB: .259
That's pretty consistently around .290-.300. Last season looks like the outlier.
Hill has a great breaking ball and you may think he gets dribblers and pop-ups more than the average pitcher, but the numbers don't show this. The numbers show he's an extreme fly ball pitcher with a very low GB-rate (36.4% heading into today), hence the home run problem that he has (giving up all those fly balls, many will go over the wall).
Hill in 2007:
Through June 7th: 5-4, 2.71 ERA, 0.98 WHIP
His peripherals: 8.4 K/9, 2.8 BB/9, 3.0 K/BB, 1.2 HR/9, .213 BABIP
Since June 7th: 0-2, 7.41 ERA, 1.69 WHIP
His peripherals: 8.1 K/9, 2.6 BB/9, 3.1 K/BB, 2.6 HR/9, .360 BABIP
He's striking out nearly the same number of hitter and walking slightly fewer, keeping his K/BB almost exactly the same, but the HR-rate and BABIP tell the rest of the story. He's basically been the same pitcher when you look at the things he controls the most (strikeouts and walks). He wasn't as good as he looked before June 7th and isn't as bad as he's looked since.
Roger Angell: I was talking with Bob Gibson and I said: 'Are you always this competitive?' He said: 'Oh, I think so. I got a three-year old daughter, and I've played about 500 games of tic-tac-toe with her and she hasn't beat me yet.'