All the Stability Control can do a apply braking to one or more wheels to limit wheel slip. Your car was already out of control and beyond help of that sort.Why didn't the stability system kick in? Moreover, after the car came to a complete stop, the engine was off. Why would the engine cuts off? Is that a safety feature? Don't I need the engine to remain ON so that the Stabilitrack will work?
If your tires are so bad that the Traction Control is annoying enough that you habitually turn it off, replace them.