If you eliminate p-value then you can't have authors that search for anything with p<0.05 and then publish, there will simply have to be some other justification. If p-value is gone it will have to be replaced with something and that something, the supposition is, will result in better science.
Writing a paper, you need to support your conclusion, removing p-value doesn't remove that need for support, it will just find something different, hopefully better.
For some context in the article they do not call for abolishing p-values but for a stop of the "significant" false dichotomy. An example would be to explain the consequences of all the values in the confidence interval or even to simply reformulate a sentence from "no significant effect was found" to "our data neither prove or disprove the presence of a significant effect"
Writing a paper, you need to support your conclusion, removing p-value doesn't remove that need for support, it will just find something different, hopefully better.