I think that you just answered your own question about why any insurer -- credit card or personal -- would care.

I suppose that the argument an insurer could make would be that since you obtained the vehicle by false pretenses, you were not authorized to drive it, and therefore insurance would not be applicable. Whether that argument would hold water I do not know. (I suspect that your personal car insurance would not cover you if you stole a car and had an accident while driving it.)