I think you are missing the point of the error detecting digits.

All they protect against is mistyping (or in the old days, misspeaking over a noisy telephone) the annoyingly long sequence of random digits. Actually judging that the numbers refer to a valid credit card account that the user is authorized to charge is an entirely different (and harder) problem.

The CC# is just a CC#, albeit one that contains extra digits that depend on the other digits.