Just download the Clinometer app which allows calibration on ALL sides of the iPhone 5S and do the test again.
I've found my iPhone 5S to be really accurate after calibration. It's the raw data from the sensors that's slightly off and it's funny that Apple forgot to allow calibration settings despite all apps that use the sensors having this.
After calibration within apps, I've found Real Racing, and Labyrinth LE works perfectly. For example, in Labyrinth LE, the ball stays put when I lay the phone flat on the table. However, Real Racing does not calibrate on all axis so if you calibrate the game with the phone standing on one of its side, you flip it around 180 degrees and the gyro sensor is off again.
Using the Compass app, my iPhone 5S is also reporting a 3 degrees discrepancy when measuring inclination standing on its side without the buttons. I tested it against the iPhone 5, all colors of the 5S/5C and the 4S at the Apple Store. It's true that the 4S and some 5C report correctly calibrated raw data when flat on its back, but none were able to give the same result when tested against all it's sides. I'm starting to believe that its impossible to have an iPhone that has correct calibration on all its sides.