They must have changed advancetrac on gen II then. My gen 1 does not apply brakes or cut engine power when you turn it off and it goes off with a single button press. It will briefly say advancetrac on or off in the message center if you have one. The light only comes on in the dash when the system is on and actively making adjustments to correct what it thinks is the car losing control. IIRC it measures throttle input, steering input, all 4 wheel speeds, and possibly some others.
When you say it won't accelerate out of a corner, is it spinning the inside rear wheel? With an open differential, you won't accelerate if your spinning one of the tires. If the gen II advancetrac is different as you described and disables stability functions with a single button press, but still leaves traction functions (braking applied to a spinning wheel) enabled, then this would help you in that situation. By applying braking to that one spinning wheel, the differential then will power the other wheel. However, I'm not sure if the system will still perform these traction functions without cutting engine power, or if it even works in that two-stage mode at all.