As 9mmepiphany has already pointed out, the eye and the gun move together. Once the target is identified, it is tracked with the body. The eye is on the target with the the eye, gun and body all moving together as a unit to track the target.
Sorry to be so late to respond, here. Computer issues. I don't see anything you say here doesn't also apply to handguns. You first identify the target with your eyes. Then you bring your gun to target. If/when the target is moving, you track it with gun and upper body as a unit, just like a shotgun. You don't move just your head and swivel your wrist.
I get it that a long gun has a stock. That makes it steadier and more accurate. But that isn't why you magically don't need sights on a shotgun. You don't need sights because the 16+ inch barrel and/or rail acts as the sights. When your head/eye is not in the right position, you'll be able to tell the barrel is crooked. Putting sights on the shotgun would just slow you down, if you focused on them. (Similarly, I can put more bullets on target faster with a Glock with open sights than I can with an AR with peep sights.)
The exact same mechanics can be applied to handguns. Once I have assumed my stance/grip, it's like I've shouldered a long gun. At this point, I've already aimed the gun. Then I move my head down and to the right, to align with the sights, just like putting my head on the stock of a shotgun. Now I adjust my aim, using my whole upper body and head as a unit.
I do see a lot of really good shooters who appear to not move their head at all, which implies they're manipulating the gun in order to simultaneously point the sights between their eye and the target, so maybe that's one of the differences between these two targeting philosphies. It would be very difficult to shoot like that without focusing on the sights.
If I focus on the sights too much, I don't get off the shots as fast on a moving target and/or between fast transitions. I'd rather get a decent shot off at the perfect time than to miss the perfect shot, entirely. In this situation, you are looking for a compromise that gives you the best opportunity for a hit (this might be another way of saying shooting a flying bird is an act of faith), and perfect sight picture is not always the best way to go. Whenever you finally get your perfect sight picture, the target might not be behind it.
I never had trouble picking up shotguns after years of handgun and rifle. First time I shot trap, I made 9 of ten. The only miss was because the setup used two shotguns. I broke all 5 on the first gun, but the pattern was off on the second one. After the first miss, I instinctively corrected, about an inch to the left. Just a guess, really. When the next clay broke, I kept that sight picture and broke the rest.
Perhaps people who only shoot pistol focusing on the sights find it difficult to learn shotgun.