Jim, this is what we do today, was a different approach applied in the muley days? I believe that is NH's question/quiz. I do think it was approached differently, how different I am not sure. We will see.
Centripetal force is the new term applied to what we used to call centrifugal force. Centrifugal force is not found in todays high school Physics book, it has been so since the late 1990's.
Centripetal force is as you said a force toward the center, and I like the "violent" you through in there. It is attached to the center by a string, chain or gravity, these are the connections to the center hence the force. This is Newtons second law F=ma. If this connection is severed or released the object will travel in a direction away from the center in a straight line, it will no longer circle the center, there for no centrifugal force exist, it is now just accelerating. The force is connected to the center, centripetal.