Finally someone who gets the correct answer.
I hate to disagree with the good counselor and Mr. Richard here, who both know much more about most things gun than I do, but that's hogwash, an old wive's tale.
I've shot straight hand double trigger guns for years, and never once during that time have I slid my hand around to make the rear trigger. That's thousands of targets and more birds than I care to admit.
The index finger merely slides off the front trigger onto the back trigger. Very fast and efficient. Changing the grip on the gun while shooting would be anathema to good form and effective shooting. I can't even imagine trying to do it!
The straight grip gun as others have indicated is better suited for fast, very instinctive shooting, for the lack of a better term "snap" shooting. The small, straight grip, which puts the rear hand at something of a less comfortable angle than a pistol grip, helps to prevent a death grip with the rear hand and forces the leading hand to take the dominant role in the mount and swing--the forward hand is what drives the gun, points it, and kills birds. The straight grip also lines the two hands more along the same plane as the barrels.
Target shooting, and some forms of pass shooting, are more a deliberate, precision game, particularly in the US where most shooters premount. Look at the exaggerated rakes on the pistol grips of dedicated skeet, SC, and especially trap guns. The pistol grip is much better suited for this kind of shooting. Like others have said it also helps tame recoil to a degree.
When I shoot my SxS at skeet or SC, my score actually goes down when I pre-mount--when I shoot my over/under or my Beretta autoloader with pistol grips the score goes up when pre-mounted.
The late writer Michael McIntosh devotes a chapter in one of his books to the different grips and the advantages and disadvantages of each. Reading it confirmed why for years I preferred my straight gripped guns for the kind of bird shooting I mostly do (grouse and woodcock). Sebago Son got it exactly right.