A man names Buzz Fawcett once had a shooting school out in Idaho, or Montana, where he taught a "low mount" way of shooting shotguns accurately at both clays and game birds. Basically, the shooter did not have his face against the stock, but instead held the gun to his shoulder's pocket, and looked over the barrel a few inches. The front sight, if there was one, appeared in the lower peripheral vision of the shooter, and was simply a reference for windage side-to-side alignment) purposes. Buzz retired from his business a couple of years ago, and I have not seen an ad for the school since. He was working at one time to write a book on his shooting method, and may still finish it. I hope he does, as I have wanted to know more about it.
Years ago, A Hollywood stuntman, and shooting coach to the stars took some stars like Jerry Lewis, and Buddy Hackett out into the desert and used this same technique to teach the students how to hit accurately at long range, shooting rifles, without either a scope sight, or iron sights. Buddy Hackett told John Carson- a very Anti-gun Hollywood personality --- about it on his Tonight Show one nite as Johnny's guest. Its one of the few times that Johnny didn't have anything funny to say to his guest. Buddy told him that they were hitting small targets out at 300 yds, with their rifles at the end of the "clinic", shooting off-hand.
So, its possible to learn to shoot moving targets with iron sights, if you learn to look over the top of them. Oh, I read an article by someone who interviewed The late Ed McGivern, the famous Pistol and Revolver shooter, who wrote his seminal book, Fast and Fancy Pistol Shooting. They asked him how he could possibly use the sights on his revolvers and pistols to shoot multiple clay ( and other aerial) targets. Ed indicated that he did not have the time to align the front blade in the notch of the rear sight, but used them as a base reference to align the gun in his hand before the targets were thrown. Once he had established his grip, he looked OVER THE SIGHTS- not at them, as he fired his guns to hit the flying targets.
I have shot thrown clay targets with my .50 caliber rifle shooting shot loads, and found I could only do it Looking OVER the sights- not at them. If I tried to align the sights, I missed the target every time. :hmm: