Updated to iOS 7.0.3 and now my iPhone 5S reads 1º off for surface, 1º off for horizontal, and 0º off on vertical (spot on). This is a definite improvement. I am fairly confident of these results as I used equipment from my physics lab.
Do I wish all three were 0º? Yes. Granted, it's a phone. But, 1º off on level over the length of an 8 feet 2x4 means one end is about 1.7 inches higher than the other. So, no home building with my iPhone…or leveling a moderate size picture. :-)
Using iHandy Carpenter or Clinometer, the iPhone is spot on. This leads me to believe that the calibration method used is all important. Both iHandy Carpenter and Clinometer rely on known perfectly horizontal and vertical surfaces to calibrate. This is not totally convenient, but it allows for spot on calibration. The native level app uses the "rotate the ball" method. This is clearly more convenient, but not as reliable. I understand Apple's aversion to using a calibration method like that used for iHandy and Clinometer. Those are not "It just Works" methodologies. You have to find, or create, a perfectly horizontal and vertical surface. But, it would be nice to have the option to use an OS wide calibration method like those two apps for those who desire greater accuracy.
Or…as a compromise, Apple could use a calibration method employed by many digital levels. This involves calibrating by taking readings of horizontal and vertical with the instrument in two orientations-180º different from each other. This removes the need for a perfectly horizontal and vertical surface.