You are looking for nice exemples of flaws in hypothesis testing. As my area of specialization is psychology, nemely develomental psychology, I use this domain to substantiate my answer to your question. In psychology it is often the case of experiments that make no sense at all. This is so, because they pretend to test what is true by defintion and hence what is untestable and not subject to falsification in Popperian terms. Think, for exemple, of an experiment designed to test that formal operations in Piagetian terms appear only after concrete operations. Because formal operations are operations on concrete operations, then the former (formal operations) cannot but appear after the latter (concrete operations). This would be a flawed experiment because it intends to test what is untestable and true by defintion. This would be also a pure waste of time. This also means that when we perform research we should be first, say, "students of logic" and only after that "students of nature". There is a paper pulished in 1978 by Paul Meehl in the Journal of Consulting and Clinical Psychology and titlel, Tabular asterisks and theoretical risks: Sir Karl, Sir Ronald, and the slow progress of soft psychogoly. I think that it is a wonderfulp paper on the flaws of hypothesis testing. I am sure that you would profit a lot from reading it.
I hope that I has got your question and that this helps.
Please let me know if this reference/site is helpful to you:
Problems with the Hypothesis Testing Approach
https://sites.warnercnr.colostate.edu/gwhite/wp-
content/uploads/sites/73/.../testing.pdfCached
of the problems with null hypothesis testing are given below (several more ... The most glaring problem with the use of hypothesis testing is that nearly all null ...