Depending on how narrowly the point is pinpointed the dating can be spread out, but it is old, some of it predates the official introduction of NHST by Fisher, see Nickerson, Null Hypothesis Significance Testing: A Review of an Old and Continuing Controversy:
"Criticism of the method, which essentially began with the introduction of the technique (Pearce, 1992), has waxed and waned over the years; it has been intense in the recent past. Apparently, controversy regarding the idea of NHST more generally extends back more than two and a half centuries (Hacking, 1965).".
That "statistical significance" is misleading as to significance in the usual sense of the word was spelled out e. g. by Eysenck in 1960:
"Eysenck (1960) made a case for not using the term significance in reporting the results of research. C. A. Clark (1963) argued that statistical significance tests do not provide the information scientists need and that the null hypothesis is not a sound basis for statistical investigation."
The more recent flare ups are associated with APA's near banishment of $p$-values in 1999 (some psychology journals did banish them), see Hypothesis testing: Fisher vs. Popper vs. Bayes, and the 2010-2014 unrest that culminated in Nuzzo's 2014 article in Nature that became one of the most highly viewed in its history. Along with spreading sentiments, also captured by Siegfried's contemporaneous quip "statistical techniques for testing hypotheses… have more flaws than Facebook’s privacy policies", it prompted an unprecedented ASA's 2016 policy statement on $p$-values, where principle #5 reads:"A $p$-value, or statistical significance, does not measure the size of an effect or the importance of a result".
One explanation as to why the point has to be made over and over again can be found in Leek's post subtitled why the $p$-value bashers just don't get it:
"Despite their flaws, from a practical perspective it is and oversimplification to point to the use of P-values as the critical flaw in scientific practice. The problem is not that people use P-values poorly it is that the vast majority of data analysis is not performed by people properly trained to perform data analysis... By scientific standards, the growth of data came on at a breakneck pace. Over a period of about 40 years we went from a scenario where data was measured in bytes to terabytes in almost every discipline. Training programs haven’t adapted to this new era... Since most people performing data analysis are not statisticians there is a lot of room for error in the application of statistical methods. This error is magnified enormously when naive analysts are given too many “researcher degrees of freedom”.
[...] P-values can and are misinterpreted, misused, and abused both by naive analysts and by statisticians. Sometimes these problems are due to statistical naiveté, sometimes they are due to wishful thinking and career pressure, and sometimes they are malicious. The reason is that P-values are complicated and require training to understand. Critics of the P-value argue in favor of a large number of the procedures to be used in place of P-values. But when considering the scale at which the methods must be used to address the demands of the current data rich world, many alternatives would result in similar flaws. This is in no way proves the use of P-values is a good idea, but it does prove that coming up with an alternative is hard."
