The reason we believe the electron to be an elementary particle is precisely because people do high precision tests like the one you reference, looking for some kind of sub-structure which would indicate that the electron is made up of more fundamental bits, and come up empty-handed.
Of course, such tests can never definitively rule out an EDM or anything like that. Instead, they place ever-smaller upper bounds on how big it could possibly be (otherwise, they'd have seen it). We're not down to the level predicted by the standard model yet (which predicts an EDM about nine or ten orders of magnitude below the sensitivity of this experiment), but this does place a limit on the influence of non-standard model physics.
Experiments like that are constantly being performed to search for evidence that our current understanding of the universe is flawed. This practice - actively searching for evidence of the inadequacies of our current theories - is a crucial hallmark of science. It's a win-win, because either the results provide further evidence for what we currently think, or they point they way toward new discoveries.