You ARE missing it. [8D]
StabiliTrack is an on road safety system that uses youe anti-lock brak system to apply brake to a sliding spining tire.
Example: You are going 55 mph on an ice road, you enter a left turn and your right front tire begins to slip, StabiliTrack senses it, and applies brake to the rear left tire to compensate and stabilize the vehicle to help prevent loss of control. This all happens very quickly and can move front to back, side to side to keep the vehicle traveling in a constant direction under control.
Traction control is easier, if a tire begins to slip while accelerating on say ice, the anti-lock brake is applied to that/those slipping tire(s) to STOP them from slipping.
Deep snow is very often handled similar to deep sand (and it is entirely different from ice and hard pack covered roads).... you WANT tire spin to dig to traction and clean out the treads of the tire.... fling that snow man!!! You WANT to control how much tire spin with the long pedal, not some goofy idiot helping anti-lock brake system. In deep snow you are not going to be going 55mph and have to hang a left turn. If you are in deep snow and you need more throttle to get you moving and keep momentum... just like sand, you do NOT want TC or StabiliTrack applying brake to the wheels, requiring more throttle, and more brake.... see the vicious cycle you are avoiding by turning it off??
On a road with a couple inches of snow, you might need the StabiliTrack to help you stay on the road and around those corners.
Your push and pull observation are right on. We want our trucks to pull us through the deepsnow.