From the linked Nature piece:
Jan de Ruiter, a cognitive scientist at Bielefeld University in Germany, tweeted: “NHST [null hypothesis significance testing] is really problematic”, but added that banning all inferential statistics is “throwing away the baby with the p-value”.
If the objective is to prevent P values from being used as stand-ins for real, repeatable results, why not require some additional significance testing in addition to disproving the null hypothesis? Psychology may have some specific issues* (i.e., small sample sizes, variation across samples, or even good ol' false positives) but much like other areas of study it requires methods to compare experimental results.
There's a temptation to suggest that reviewers should be responsible for determining whether a P value is used appropriately. A seemingly-significant P value has great power, though, and the average reviewer may not read beyond P < 0.05 even if other analyses are presented or even failed. As with nearly every other aspect of publishing, results and their interpretation should be handled on a case-by-case basis without eliminating any options. Abuse of statistics will only get worse without consistent ways to interpret data.
* I'm not a psychologist, have never published in a psychology journal, and rarely speak with anyone practicing psychology research.