I have heard these arguments before, but my experience in skeet shooting refutes them. In shooting doubles at skeet, the targets are going in opposite directions and having the gun shouldered has absolutely no effect on acquiring the second target. Moreover, suppose you should notice that the case has failed to eject. I can assure you that at that point it is too late to do anything about it. In skeet shooting you would be allowed another shot, but not in dangerous game hunting.

In shooting international skeet, where the target is moving in the range of 65 mph (as opposed to 50 mph in American skeet) plus the shooter must start at the low gun position and is not allowed to start raising his gun until the target is out of the house, if the shooter took the time to unshoulder and reshoulder the gun, the second target would already be out of bounds before he acquired it. Moreover, since there is up to a three second delay in releasing the target after it is called for, the shooter must respond to the target, not the other way around, reducing even further the time available for the first shot. Given the speed of the target (roughly 95 fps.) and the distance to "out of bounds" (40.3 meters or 132 feet), the shooter has only about 1.5 seconds, minus reaction time, to shoot both targets.

Most experienced skeet shooters, like most experienced rifle shooters, shoot with both eyes open, so the only blockage of the field of view is that caused by the gun barrel. Unlike skeet shooting, shifting targets is not involved, except in the unlikely event of being charged by more than one animal at the same time. I once was involved with a pack of Cape hunting dogs who had just made a kill on fenced in private property. I emptied my Model 70 .300 H&H at them and killed four with five shots, all but the first one running shots, and I can assure you that I did not unmount and remount my rifle between shots.