"In addition, Political Analysis will no longer be reporting p-values in regression tables or elsewhere. There are many principled reasons for this change—most notably that in isolation a p-value simply does not give adequate evidence in support of a given model or the associated hypotheses. There is an extremely large, and at times self-reflective, literature in support of that statement dating back to 1962. I [Jeff Gill] could fill all of the pages of this issue with citations. Readers of Political Analysis have surely read the recent American Statistical Association report on the use and misuse of p-values, and are aware of the resulting public discussion. The key problem from a journal’s perspective is that p-values are often used as an acceptance threshold leading to publication bias. This in turn promotes the poisonous practice of model mining by researchers. Furthermore, there is evidence that a large number of social scientists misunderstand p-values in general and consider them a key form of scientific reasoning. I hope other respected journals in the field follow our lead."
This is misguided in that it identifies a symptom of bad research as the problem itself. I think that many conclusions based on models published in political science are unreliable because the social world is complex and hard to measure. Yes, evaluating support for theory based on whether a coefficient in a model is statistically significant or not is often a quite poor way to answer this type of question, as it depends on the model and the assumptions of the variance estimator, but banning p-values doesn't solve this problem, just treats (poorly) a symptom. Note that Political Analysis didn't ban confidence intervals. These are just inverted hypothesis tests!
"Model mining" (which reminded me of Ted Cruz calling basketball hoops "basketball rings") is not inherently bad and should not be discouraged. It should be made explicit, and its statistical properties studied. This is an entire field/class of methods (machine/statistical learning) that have been casually dismissed. Yes, doing model selection via an informal search for small $p$-values for certain "effect" estimates has poor statistical properties. That isn't new though.
Since the original editorial PA tweeted
"Important clarification to PA's recently updated p-value policy: in the case of design based experiments and related procedures (permutation tests, asymptotic approximations, etc.) p-values are appropriate and may be supplied."
Aren't we nearly always making an asymptotic approximation to the sampling distribution of a statistic?
p-values are quite sensible when used appropriately. If the quality of our methods is poor then our reviewers need technical help and our students need better education. Banning p-values is not a solution to these problems. They can be a sensible tool.