One more thing about BA, and you've brought this up about fielding percentage, too. Who calls an error an error? A subjective person, so if one player gets a hit on a certain play, but another gets an error on the exact same play, their BAs would be skewed.
Now, it all is supposed to work out over a long period of time, but errors are judged by the home team's scorer, not some unbiased person, so some players might be helped or hurt over the long run.
Of course, one could and should argue the observers used in UZR/150 are subjective and biased, too, and they'd be right, but at least they are trained and calibrated and rotated.