So, to clarify, we should remember that there are two different systems at work here - ASR ("Traction control") and VDC ("Stability control"). They both have the same yellow light to show when they are working, but they are two very-different things. I think people confuse one with the other, since they have the same light.
VDC prevents the car going sideways. It can't be turned off. And actually, you don't want to turn it off. I drove my Stilo Abarth on a track once, and I can tell you that by the time VDC intervenes, you are well out of shape (past the region of good handling for optimal lap times) anyway. I only managed to get VDC to intervene by clumsy hard braking while turning into a corner, 'flicking' the steering while snapping the throttle closed, etc. It was absolutely impossible to spin the car. Remember that on a track, 'sideways' is not the fastest with a front-wheel-drive car. There is no advantage to being able to get the car sideways. Therefore, there is no disadvantage to having VDC 'on' at all times.
However, I hope I've also explained that having VDC on does not help the handling under good conditions, since it intervenes at a very late stage as a 'safety net' only. That is why it isn't fitted to race cars as an 'unfair advantage'
ASR can be turned off. I did five standing starts with it on, and five with it off, and there was no discernible difference in the times. Obviously, with it off there was lots of wheelspin even on the dry track. With it on, there was no wheelspin. Simple as that. gotcha makes a good point there when he describes the emulation of the LSD. Again, since there's no advantage to turning it off, why not leave it on?
At the end of the day the effectiveness of the car depends on the level of grip and traction available. Although Tiff shows that electronic systems make the most of what is available, there will never be the same level of grip and the same traction on ice as there is on a dry road. I'd dispute that Tiff's stability test really is on 'sheet ice', as there seems a lot of snow on the surface. Compare it to Jeremy Clarkson's ice-skating in YouTube - Jaguar XK vs. speed skater - Top Gear
ESP/VDC wouldn't be much help there!
The answer to this thread, therefore, is "yes, but don't expect it to be as good as on a dry road".