ok, i will give this a shot (pun intended)
there is a lot of nostalgia involved here, but having said that it is also about your shooting style, weather you are shooting aimed (O/U) style or if you are good enough to used the advantage of instictive shooting (S/S).
Aimed shooting:
Standing on a peg(either at a driven shoot or clay shooting) you generally have a good indication from where the targets are coming from and have all the time in the world to aquire and follow through on that target, most people can learn this and get decent results, but when they are with the cocker spaniels in the scrup after woodcock, the speed needed is most likely lacking
this type of shooting is mostly a learned type of shooting, which means that it is all in your head, and your motor reflexes are controlled by your brain( concius trigger pull)
Instinctive shooting:
Peg shooting in close quarters(only getting short glimpses of your targets) or moving with your gun after the dogs in brush, the shooting lends itself more to instinctive shooting where your gun moves as part of your body and the first shot usually is shot as soon as the butt of the gun hits the shoulder, this takes practise and more than a bit of instruction from a instructor that knows how to teach as well (most instructors can be very good at pegs and know nothing about instinctive shooting and therefore denies its qualitys

)
when mastred this tecnique can be transfered to your double rifle making you one of the fastest shots on the drive, and making it possible to actually get that leopard before it gets you
one of the reasons why most competitions are done with O/U are in my oppinion, that the O/U is the first gun most people get in their hand at the shooting range, because they are less expencive and a bit heavy as compared to the S/S game gun, so you can actually shoot a lot without getting hurt, and when people start out they cant afford a well balanced S/S so they actually learn the aimed tecnique, and it is a lot harder to go from that to instinctive shooting, so sadly most people never progress to the second level.
just my oppinion, and that is worth what you payed for it
best
peter