- The OP was a bit "loose" with words when writing "When the sample gets large, the data will be approximately normally distributed." What should have been said is that "When the sample gets large, the sampling distribution of the means of the samples will be approximately normally distributed."
When using a t-test e.g., we do not care about the original distribution being normal, we care about the sampling distribution of the statiustic of interest being normal... So, yes, with large samples, parametric tests of the mean are more valid/powerful than any non-parametric test.
Several commentators jumped on this a bit too fast, as the basic question remains valid (is there any valid practical use for the Wilcoxon Signed Rank test?)
- The Wilcoxon signed rank test (WSRt from now on) does not test the mean, or the median, as is regretably too often written.
The WSRt instead tests the pseudo-median (https://en.wikipedia.org/wiki/Pseudomedian). When the sample is symetric, then the pseudo-median is equal to the median (and the mean). But samples are NEVER absolutely symetric, so...
The best wording for the null of the WSRt that I could find is that "the data is symmetrical around the hypothesized target".
- The problem with that null is that, when you get a significant result, you have no idea what it means. It could be because the data was not symmetrical, because it was symmetrical but around a median different enough from the hypothesized one, or (basically all the time: all sample data will have some non-symmetry) because of a combination of both.
- Note that when the data is symmetrical "enough", then a) mean=median b) CLT converges very fast because your sample is not very "non-normal".
- If data is "very non-normal", then the data is also "very non-symmetrical". So WSRt is not applicable.
- Everywhere you could use a WSRt, you could use a Sign test (which does not rely on any assumption: and if the sample is large it will have power). Or even a Kruskal-Wallis U test (KWUt), even for a single sample (construct an artificial sample of the same size, with all values equal to your hypothesized target). And at least there is an intuitive interpretation of significance of the Sign test (median) or KWUt (stochastic dominance).
So, after all this, I have reached the conclusion that the WSRt is basically useless, at least in applied statistics (it may have some use in mathematical/theoretical statistics), because the pseudo-median has no intuitive interpretation, the symmetry assumption is untestable and the test is not robust to departures from symmetry, while a t-test is robust (with respect to type 1 errors) to departures from normality, and has intuitive interpretations for significance. From all I can tell, it is, at best, a historical relic.
And based on the answers in your post, no-one (yet?) presented a possible business case for the WSRT. QFD...