What's your definition of core stability? I believe it is the ability of the upper body and lower body to act independently when necessary and to work co-dependently when necessary. There's a degree of autonomy per quarter of the body, and a (much larger) degree of harmony between them as well. This balance or communication, I believe, is the essence of stability.
Think about throwing a baseball: the pelvis and rib cage must rotate away from one another during the wind up, but once the stride (lead) foot contacts the ground, the throwing arm gains stability from the lead leg and their linkage contributes to throwing performance. When executed properly, complex motions like throwing a baseball or the golf swing are the pinnacle demonstrations of "core stability". Everything somehow moves in opposite directions, yet is very much connected still.
I believe this is the brilliance behind exercises like chops and lifts. They challenge both linkage and separation in stance-forms that are the foundation of complex movement. Another exercise that deserves more recognition among HARDCORE strength and conditioning professionals is the "Dying Bug".
Before I go further, read this article and this article from Craig Liebenson, DC. I hope you find, as I did, the power of the "Dying Bug" position not only for rehabilitative purpose, but for movement purpose. If you are a coach who is still perturbed with the idea of developmental kinesiology, I recommend you read the first 3-4 chapters in these two books:
In the Dying Bug variations, you challenge centration (proper alignment) of the rib cage and pelvis with different limb lowering and twisting combinations. The exercises can be progressed from low-intensity, so much that breathing can be maintained throughout, to high intensity where a deep breath is taken and held under a braced abdomen. Recruitment of the pelvic floor ("hold in your pee") should be demanded throughout.
In the video below, I demonstrate some progressions in the supine, Dying Bug position. The intensity of the exercises is hardly communicated through video; however you do see my right hip integrity get insulted by trunk rotation to my left. Contrary to previous estimates, I'm not a robot.