My theory is the cause of snap-oversteer is lack of feel in most NSXs (I'm excluding cars with race-worthy suspension on top of slicks, etc). In the video posted the driver literally prevented the wheel from turning to the right quick enough to get back to where it needed to be (which it would have done for him at that point if he had simply let go). Now, letting go is not the thing to be doing at high levels of performance driving, racing, etc. as drivers can certainly do better than what a vehicle wants to do naturally. However, for many drivers what the car wants to do (what it would do if they let go) is better than what they would do. it is also not a bad way to learn feel. You at least see what the car wants to do...see where the steering wheel should be if the front wheels are to be pointed in the direction of the slide (momentum of the vehicle).
That basic knowledge/understanding can then be built upon...and folks can learn to improve on these natural dynamics (anticipation, smoothing things out, etc). I suppose in certain conditions snap-oversteer might be a natural thing for an NSX...but most of the examples I've seen (the posted video, at track events, etc) it is because folks over-react or are just too slow on secondary reaction (don't anticipate recovery, etc).
I very much recommend driving on dirt (other cars), ice, snow, rain, etc as building blocks to having such feel even in the dry (where things happen much more quickly). Sounds like you're doing some of what may eventually contribute to having feel (autox, drifting, etc).