Hypothesis testing such as Anderson-Darling or Shapiro-Wilk's test check normality of a distribution. However, if the sample size is very large, the test is extremely "accurate" but practically useless because the confidence interval is too small. They will always reject the null, even if the distribution is reasonably normal enough.
How should I test normality when sample size is very large, other than visualizing histograms?
The motivation is that I want to automate checking normality of large data set in a software platform, where everything needs to be automated, not manually visualized and inspected by humans.
One thing that came across me is that instead of using Shapiro-Wilk test, I calculate kurtosis and skewness of the distribution, and if they are $\pm 1.0$, I can assume that my large dataset is "reasonably" normally distributed.
Is my approach correct, or is there any other alternatives?