To ban or not to ban the P-value: A question of qualitative vs. quantitative data raised by ASA and BASP

TJ posted an article earlier this month about how the ASA issued a statement concerning the P value, saying that "statistical techniques for testing hypotheses.... have more flaws than Facebook's privacy policies."

I found out after some click-holing from article to article about statistics and the P value that Basic and Applied Social Psychology (BASP) has actually banned the P value since 2015, and more specifically, the null hypothesis significance testing procedure, or NHSTP. BASP even states that prior to publication, all "vestiges of the NHSTP (p-values, t-values, F-values, statements about 'significant' differences or lack thereof')" would have to be removed. And this basis arises from the fact that numbers are being generated where none exist, a problem that I feel is more specific to more qualitative fields such as psychology. But the BASP raised the same concerns as the ASA statement regarding the p value: that p < 0.05 is "too easy" to pass and sometimes "serves as an excuse for lower quality research." While the ASA's concern is mainly geared towards the people and scientists who perform the research (i.e. people are not properly trained to perform data analysis), it seems that BASP's concern arises from the nature of psychological research, stating that "banning the NHSTP will have the effect of increasing the quality of submitted manuscripts by liberating authors from the stultified structure of NHSTP," even stating that it hopes other journals will follow suit.

I certainly understand where BASP is coming from - with a field where response/measured variables are more often qualitative than not, how does one effectively apply statistics to analyze whether or not an effect is real? What should we do about data generated in the "hard sciences" that are more qualitative, such as characterization of cell morphology? What about clinical data that measure subjective things such as level of pain on a scale of 1-10? Is there an existing statistical tool or procedure out there that everyone could agree would accurately measure "significance" without having to apply values or generate numbers to describe qualitative measurements?

Do you think we should abolish P-value significance testing for all research? Only psychology research? How about all qualitative vs. quantitative research?