Looks like you and your son had a blast out there!
In regards to your question about the TC coming on at turn exit; most (all?) car mfg's believe that understeer is safer than oversteer so reducing the power to the rear wheels is intended to keep the rear wheels from losing traction and the back end sliding. In your case, the TC is simply helping to keep the rear end behind the front end (all it takes is a slight nudge of the rear brake). If the driver really overcooks the corner, TC keeps the car manageable while the front end understeers off the track into the weeds.
The difference here is that the TC has the ability to only apply the rear brake to keep the rear end from sliding. If you as a driver were to tap the brakes in the same situation, then the weight would shift forward and the rear end would slide even more.
With that being said, your intuition is correct - if the rear end starts to slide an experienced driver would want to keep the weight on the rear (or increase it) by keeping the throttle pinned and either give it a little counter-steer or unwinding the steering a bit to open up the corner.
Eventually you'll find that you are engaging the TC on a near-continual basis and that it is actually slowing you down.
Its great that you were able to ride along with an instructor, that is always worthwhile and will help set your calibration point for what the car can really do.