The better made doubles have some "cast" in the stocks that makes them fit most shooters better, especially as Americans get heavier and have fatter cheeks. Pumps have an inherent tendency to line up faster for the second shot because of the pumping action, if you keep your eye on the bird.
Some snobs claim doubles are the "most sporting" guns to shoot because you only get two shots. If you think that is true, just put two shells in a pump and you'll have something more "sporting" than a double, because the pump is inherently more difficult to shoot twice.
You'll rarely see anyone shooting well with a pump in competition because even the best pump shooters will occasionally "short shuck" when the pressure of competition is on.
English gunsmiths promulgated the snob appeal and myths about doubles being best so they could justify the higher prices they were charging the aristocracy, although the classic early doubles truly are works of art.
If you want to see what guns are "best" if best means easiest to shoot and hit the target, start going to sporting clay competitions and notice what all the best shooters are using. They even claim that over and unders are the best of the doubles because there are certain angles where targets are blocked out for an instant by the wider side by sides.