So let's accept the that the "statistical significance" label has some severe problems, as Wasserstein, Schirm, and Lazar write:
[A] label of statistical significance does not mean or imply that an association or effect is highly probable, real, true, or important. Nor does a label of statistical nonsignificance lead to the association or effect being improbable, absent, false, or unimportant. Yet the dichotomization into “significant” and “not significant” is taken as an imprimatur of authority on these characteristics. In a world without bright lines, on the other hand, it becomes untenable to assert dramatic differences in interpretation from inconsequential differences in estimates. As Gelman and Stern (2006) famously observed, the difference between “significant” and “not significant” is not itself statistically significant.
But as they recognize, criticizing is the easy part.
What is to be done instead? And here, the argument fragments substantially. Did I mention that there were 43 different responses in this issue of the American Statistician? Some of the recommendations are more a matter of temperament than of specific statistical tests. As Wasserstein, Schirm, and Lazar emphasize, many of the authors offer advice that can be summarized in about seven words: "Accept uncertainty. Be thoughtful, open, and modest.” This is good advice! But a researcher struggling to get a paper published might be forgiven for feeling that it lacks specificity.