Handy may have been thinking that this was a topic where responses needed to be dumbed-down a little (3800 posts will do that to ya).
The active ingredients in most toothpaste are flourides & peroxides. Handy was probably referring to the abrasives that some toothpastes contain. Active ingredients are those that will, by themselves & without any externally applied force or energey, act against or react with another compound or entity. The abrasive surfactants (“sand”), while effective as cleaning agents, are not by themselves “active” (or even effective) unless forced into action (toothpaste by a tooth brush, for example).
Alcohol, for example, usually shows up under the inactive or other ingredients list even though it is clearly an active chemical compound.
It’s nice to hear that your sweater & CDs were rejuvinated with a little dab o’ crest, but I would only try that if the items in question were one step away from the trash can anyway. Sometimes scratches can be buffed out by using successively finer abrasives (I once successfully reclaimed a scratched magnifying lens this way) but there’s an excellent chance that you’ll do more harm than good, so be prepared to toss it in the trash if things don’t work out.