I’ve only glanced at the paper and I’m not a genetics expert but I don’t understand his comments about the MWU tests lack of applicability. I’m guessing the author is just doing a finite sample correction for the median as opposed to the mean. And I think the point is that using the finite sample correction will increase the statistical power of the test. Because if we perform a finite sample size correction for a test of the mean then the standard error will decrease and hence the power of the test will increase. I think the authors point is that we can do the same thing for the medians too, i.e. increase the power of the MWU test by using the authors finite sample correction. BTW, finite sample corrections can be done for sample size estimates too, maybe he should derive a sample size/ power analysis formulas for his population known median test thing?

https://en.wikipedia.org/wiki/Standard_error#Correction_for_finite_population

https://onlinecourses.science.psu.edu/stat414/node/264

If a whole population is measured, surely all statistics based on random sampling are rendered meaningless? The point of Mann-Whitney is the same as ANOVA, T-test etc - to determine the probability of a result recurring at least as strongly if a new sub sample is taken at random using the same selection criteria from a parent population. This is why they fall apart if bias exists in the sample selection.

Surely a test would need to check if effect sizes exceed measurement error to be relevant?

As with all statistical analysis there would also need to be an assessment of practical significance - would the effect size make a meaningful difference to anyone or anything even if statistically significant.