On 25 February 2015, the journal Basic and Applied Social Psychology issued an editorial banning $p$-values and confidence intervals from all future papers.
Specifically, they say (formatting and emphasis are mine):
[...] prior to publication, authors will have to remove all vestiges of the NHSTP [null hypothesis significance testing procedure] ($p$-values, $t$-values, $F$-values, statements about ‘‘significant’’ differences or lack thereof, and so on).
Analogous to how the NHSTP fails to provide the probability of the null hypothesis, which is needed to provide a strong case for rejecting it, confidence intervals do not provide a strong case for concluding that the population parameter of interest is likely to be within the stated interval. Therefore, confidence intervals also are banned from BASP.
[...] with respect to Bayesian procedures, we reserve the right to make case-by-case judgments, and thus Bayesian procedures are neither required nor banned from BASP.
[...] Are any inferential statistical procedures required? -- No [...] However, BASP will require strong descriptive statistics, including effect sizes.
Let us not discuss problems with and misuse of $p$-values here; there already are plenty of excellent discussions on CV that can be found by browsing the p-value tag. The critique of $p$-values often goes together with an advice to report confidence intervals for parameters of interest. For example, in this very well-argued answer @gung suggests to report effect sizes with confidence intervals around them. But this journal bans confidence intervals as well.
What are the advantages and disadvantages of such an approach to presenting data and experimental results as opposed to the "traditional" approach with $p$-values, confidence intervals, and significant/insignificant dichotomy? The reaction to this ban seems to be mostly negative; so what are the disadvantages then? American Statistical Association has even posted a brief discouraging comment on this ban, saying that "this policy may have its own negative consequences". What could these negative consequences be?
Or as @whuber suggested to put it, should this approach be advocated generally as a paradigm of quantitative research? And if not, why not?
PS. Note that my question is not about the ban itself; it is about the suggested approach. I am not asking about frequentist vs. Bayesian inference either. The Editorial is pretty negative about Bayesian methods too; so it is essentially about using statistics vs. not using statistics at all.