Without staggering, the pixels will not be exposed uniformly resulting in the last pixel being stretched out, but how do we deal with the rolling shutter artifact problem?
From my understanding, you can use a camera with a different type of sensor technology such as CCDs. There is a comparison here. The animation is probably in flash which is why it doesn't work.
Is the readout speed a limitation on the shutter speed? It seems this would be the case, as if the exposure time is less than the pixel readout time, then there's not enough time to stagger the readout so all pixels can get the same exposure time.