The total is indeed nearly normally distributed, but the rate of convergence (particularly in the tails) is not fast enough to avoid having those very sensitive tests give wrong results.
Were it otherwise there would have been no need to develop the chi-square test. It would have been entirely redundant. (It actually is redundant because we have the g-test. But evaluating the chi-square test just involves taking squares, while the g-test involves taking natural logarithms. This made the less accurate chi-square test much easier to do when people didn't have computers to calculate it on. Today we should use the g-test, but few people have heard of it.)
