There may be inferences you can make from the sum that aren't immediately obvious. If cards can end in four zeros, the sum and the last four digits contain equivalent information, but you would also confirm that three of the digits are zeros if the sum was 1. It's something that, if I were a bank, I would want someone with a background in number theory to weigh in on. If I were a paranoid bank exec, I wouldn't trust the low-wage customer support reps I had on staff to vet customer questions for how much information they might leak and would instead have blanket prohibitions on answering questions from customers until after the authentication phase of the call.

Questions like "is the sum even?" trade a lower opportunity for information leakage for a greater opportunity for a random guess to be correct.