Hitting: wRC+. This one was the easiest for me. Based on wOBA, the second best statistic IMO, it compares how well a player hits compared to league average. wRC+ is also a park and league-adjusted, allowing one to to compare players who played in different years, parks, and leagues. I love rate stats, and I love wOBA and wRC, so I love wRC+.
Pitching: FIP-. This was an excruciatingly hard one for me, but I went with FIP- over ERA-. Again, it's a rate statistic, so it compares how good a pitcher is compared to league average; the big difference between it and offensive neutralized rate stats is that the lower, the better. I also went with FIP- over FIP because it takes into account league and park factors; in FIP, pitchers in San Diego almost always have lower FIP's than those in Colorado or Yankee Stadium, all other things being equal. It was close between ERA- and FIP-, and though some sabermetrician fans hate ERA, I don't hate it; I accept it for what it is- an indication of what really happened. Still, I like to factor in future performance as well, as I'm all about adding the best players to the Phillies, and I want to strip out defense, luck, and sequencing; and I believe FIP and FIP- does this best.
Fielding: Def. Also a very hard one for me, and I admit I'm the most ignorant of defensive statistics of any type of statistic, but I like the fact that Def is based on UZR and positional adjustment. For me, this makes it much more useful than UZR/150. Also, because it factors in positional adjustment, you can compare defensive players across all different positions, something you cannot do with just UZR or ever DRS. Lastly, because UZR accounts for league and park adjustment, so does Def, making it even better.
Of course, the idea of a "catch all" statistic is absurd; this thread is just for fun. One should always survey all the data available, not just one statistic. The best evaluation of a pitcher would be looking at his ERA/FIP/xFIP, K/9, BB/9, WAR, WPA, GB%, etc. The same for hitting and defense. I just wanted to see what you guys would like in a hypothetical situation where you had to evaluate two different MLB players based on only one statistic.
If anyone has any correlational graphs to different statistics and wins, or something similar, please feel free to post them! I would find that very interesting.