On this particular point, the late Mel Siff in his works constantly emphasized the Eastern Europeans’ aversion to specific core training for these very reasons. He recounts stories of seminars given by (I believe) Angel Spassov in the US in the late '80’s where American coaches would ask him the best way to load abdominal exercises, and he would just laugh and state that the abdominal muscles contract far more forcefully during squats, cleans, etc. than during any kind of weighted situp.
If you’re going to rethink core training, it might be to drop it altogether. Of course Charlie has been an advocate of large volumes of core work, BUT you also have to understand how he uses it in the overall structure of the training and the loading parameters used, which differs greatly from many of the core training gurus pushing their gadgets and DVDs.