I am not proficient in scientific method or statistics. However, I observe that the article at the link seems to criticize use of the NHSTP based on the fact that there are widespread confusions about the actual meaning of NHSTP. One would think that the solution to this problem is to better educate people about the actual meaning of NHSTP, rather than to ban its use in a journal. However, presumably the grounds for banning NHSTP are in part due to the misplaced degree of importance it is given based on those common misinterpretations. Or something like that.
Perhaps BASP's decision to ban the use of NHSTP is at least partly symbolic--as a way to signal their commitment to more rigorous standards for assessing the importance of research findings than the typically lazy or otherwise problematic uses of the old stand-by, NHSTP. And the ban will get people's attention more than just not requiring the use of NHSTP, as the journal had previously done.
On a related note, this video contains an informative critique of the use of p-values and a defense of the use of confidence intervals as an alternative criterion for assessing the importance of research findings (if that's the right way to put it):