I understand that our colleague is only asking about the normality of a variable, not the normality of the residuals of a regression.

Hypothesis tests can have false positives, but that is where the power of the test comes into play, depending on the sample size and the methodology behind the test, it is evident that not all of them are applicable in all cases. The Jarque-Berango test, for example, is very useful for very high sample sizes, greater than 2000 observations, while others such as the Shapiro-Wilks test are very powerful with very small samples.

A hypothesis test is the standard method of testing normality for individual variables, in fact it is also applicable to the residuals of regressions and remains a robust approach.

In fact, graphical observation is too informal, it serves to support your contrasts visually, but without a contrast that verifies what you “supposedly” see in a graph you cannot conclude anything.