A few weeks ago, I got to test my brand new AS3+ tires on my P85D on a steep ice covered driveway. The driveway can't be walked on without slipping...that's how bad it is. I tried the previous week in my prius with tire chains and it was a no go. The Tesla went up without any slippage like it was just the dirt that's under the ice rather than the ice itself.
By far the most capable of the AWD and 4x4 vehicles I've owned. I think tires might make a big difference. The AS3+'s have an excellent ice and snow rating from user surveys on the tire rack. Interestingly, the non plus AS3s have a lousy rating.
After thinking about this for several days, I think this is the the issue: It has responsive torque cut, which gives it the *feeling* of being planted, since it's less likely to have human detectable spin or upset the direction of the car. This sort of works on low traction but otherwise hard surfaces. The real limitation on loose packed surfaces seems to be the inability to direct torque smoothly to each wheel left/right. Loose packed surfaces require consistent spin for optimal traction. My Subaru has 2 vicously limited (NOT viscously coupled) differentials with somewhat of a high preload factor, i.e. if you put both rear wheels in the air, it's quite difficult to turn a wheel by hand, since the opposite side of the differential tries to spin backwards but is vicously limited. This means each side of the differential has a certain amount of torque it will apply in any condition, without viscous locking. Note that the base Subaru's don't have this set of performance LSD's.
TLDR; no limited slip diff's and poor differential braking means bad performance on loose packed surfaces. No Colin McRae tributes in the Model S.