I would like to know why exactly calibration programs (usually?) ask the user to place the joystick in middle, that is: Why isn't moving the handle in a complete circle (usually?) enough to calibrate a joystick?
Moving the joystick in a complete circle gives you the upper bounds of the joystick range, but the at rest (i.e. no input) position of the joystick can't be accurately determined from those values.
You could assume that that rest position is the exact center of the available range but that is only true in ideal circumstances. Over time controllers wear in non-uniform ways. You want the actual rest position for the device, which can only be accurately determined by asking for it during calibration.
It is possible they are trying to remove the user from the calibration. Usually joysticks are spring loaded to return to the centered (neutral) position so really what they are asking users to do is to just let go and stay hands off during calibration.
The less human interaction, the more consistent the results.