One way to test various competing explanations for an identified causal effect of X on Y is to do a mediation analysis. In this paper, Hainmueller et al are trying to understand how ideology is related to Grexit, by unpacking the ideology’s effect into 4 components: redistribution attitudes, empathy, support for the EU, expected Grexit Impact. They find strong support for the effect of the last component.
It is tempting to think of a related research agenda. Suppose instead one were to test the same relationship — between ideology and Grexit — with the desire to dismiss the direct effect of ideology and propose instead that all its effect is captured by the mediator — in other words, an insignificant ADE. Could one say that ideology has no effect and instead the mediator has picked up the ideology’s effect on Y? I think no. Because of the way mediation analysis works, the aim is to find an ADE effect not to dismiss one. If ADE turns out insignificant, I think that is rather a case of “no evidence” rather than contrary evidence per se.