Yeah, there are pros and cons on both sides of this. From those blog posts it seems that the recommendations of the ASA 2016 statement are most reasonable (and I think they are emphasized in the webinar I link to):

P values can indicate how incompatible the data are with a specified statistical model.

P values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
 Scientific conclusions and business or policy decisions should not be based only on whether a p value passes a specific threshold.
 Proper inference requires full reporting and transparency. P values and related analyses should not be reported selectively. Conducting multiple analyses of the data and reporting only those with certain p values (typically those passing a significance threshold) renders the reported p values essentially uninterpretable.
 A p value, or statistical significance, does not measure the size of an effect or the importance of a result.
 By itself, a p value does not provide a good measure of evidence regarding a model or hypothesis.
I also recall reading (or hearing) that the pvalue was currently the only thing holding back a floodgate of irreproducible findings and I agree that the 2nd ASA statement calling for the removal of the pvalue goes to far.
Maybe the key point is that while frequentist statistics are mathematically sound, most people don’t really understand what significant results imply. I think point 2 above is a common problem for biologists  the assumption is that p<0.05 means there is a >95% chance of the two sampling populations being different, while it actually means there is a <5% chance of the observed difference occurring if the populations are the same.