A stirred culture without continuous aeration offers no advantage over a well-shaken culture when propagating a White Labs vial, and a well-shaken culture has fewer possible sources of contamination.
Doesn't Kai's experiment show that the stir plate helps to aerate the culture? I guess that is what I took away from the fact that the starter covered with an airlock grew less yeast than the starter covered with foil which grew less yeast than the uncovered starter.
If no aeration was being provided by the stir plate, wouldn't you expect to see the same growth regardless of how tightly the flask was covered? Maybe a repeat experiment with control cultures not on stir plates would be needed to really answer that.
Periodically shaking the starter during the first six to twelve hours can be beneficial.
If nothing else, I suppose a stir plate can do the shaking for you while you are sleeping or at work.