Allright now I'm sure there is something wrong with the sensor, I've switched the supply power of the wmp to 3.3, no difference, so I've removed the calibration of the gyro, the average value is around 8100, which makes perfect sense (half of the full-scale), the lowest value I get and for which the saturation happens is of course... 0. I've somehow managed to find one of the rare really crappy Wii Motion Plus copy.
It would have been great to confirm all of that with an other WMP but nevermind.