If shotguns are better why did police move away from them?
Actually they haven't really. I work for a large metro agency , and every car has a shotgun, and every cadet gets trained on them. Rifles require a 40 hour school, and probably less then half of patrol officers have them (even with personally purchased rifles authorized). I will agree that they aren't pulled on calls as often as rifles, but I think there are a few reasons that have nothing to do with effectiveness. Most police cadets have little practical experience with firearms in general, and little actual knowledge of how far buck can be effective and accurate. We give an extensive lecture and demonstration on shot patterning and ballistic effect, to teach our cadets how well a shotgun can work. Since we started doing that, we've had them pulled out on calls a lot more often, and used on calls a lot more often. I don't think we've had a patrol officer shoot someone with a rifle in at least 2-3 years (we've had a shotgun used at least once a year).
However #4's are more comfortable to shoot. Many police agencies use #4 instead of 00. Lighter recoil and more shot downrange improves hits probability.
I'll certainly not claim to know what all US LE agencies use for loads, but of the agencies I've spoken with (I'm a full time firearms instructor) none use anything smaller then 0 buck. I've personally wanted to do some testing on #1, which will hopefully happen this year. From all the reading and testing I've seen online #4 is very marginal in terms of penetration, which is a key factor to producing effective fight stopping hits. #1 appears to be the smallest size shot that has the requisite mass to ensure adequate penetration. As far as recoil go, proper technique can allow you to shoot slugs or buck just as quickly and easily as #7.5 birdshot.
For home defense lighter recoiling loads such as #6 accomplish the same task as 00. The distances in most homes are close probably not more than 30' (10 yards).
Again my major concern about smaller lighter rounds is the lack weight and momentum leading to low penetration.
As far as shotgun accuracy goes, at conventional HD distances certain loadings can give (and require) just as much precision as a rifle. See:
http://www.thehighroad.org/showthread.php?t=768385 for some patterning we did at work. In particular check out the Hornady Reduced Recoil and Federal Tru Flight patterns at 5 yards. 15' is going to be a good sized room in most houses, and it will easily allow you to pick an eye socket to engage into. These two loads can stretch a half head (or less) hostage shot to 10 or even 15 yards depending on how exactly it patterns in your particular barrel. Even the two worst patterning (Remington 9 pellet and S&B 9 pellet) can easily take a half head hostage shot at 5 yards.
The only way to effect a stop is through tissue destruction. An 8 pellet loading of 00 buckshot has a frontal area of approximately .688" inches. Premium defensive rounds for 5.56 platform will usually expand in the area of .5" inches. Fragmentation can dramatically increase that amount, but basically the same way buck works so well. Multiple divergent wound tracks that are more likely to strike something vital. Buckshot has an advantage in depth and width of these tracks, fragmentation gets the advantage in shear number of tracks. Slugs are an entirely different beast, and simply flatten into a 1.5" lead washer that travels about 16" in bare gel. Over penetration concerns for slugs are vastly over stated. As slugs expected have excellent terminal effect in the real world as well. Winchester produces a segmented slug that basically breaks into 3 165 gr projectiles on impact (each projectile is the weight of a .40 S&W slug), and it's moving about about 500 fps faster then a .40 S&W round is at the muzzle. Needless to say it has pretty solid terminal effect as well.
The shotgun looses out to the carbine without question in a few areas: the need for extreme precision, the need for extended engagements if slugs aren't available, the need to engage multiple targets without the opportunity to reload (though the AR realistically has the same problem), and the need to engage through soft body armor. All of these are areas your average HD scenario wont involve, nor will your average police shooting require. Forethought regarding ammunition selection can rectify pretty much everything except magazine capacity (which you can fix if you're willing to deal with the weight penalty), and the need to penetrate soft armor.
I did not grow up shooting at all, and only started shooting shortly before I got into LE. I have FAR more time on the AR platform, and probably an order of magnitude more rounds through one then a shotgun. I don't find bead sights naturally accurate, and hate cross bolt safetys (I wish we used Mossbergs). With all that said, it's not difficult to learn to run a pump shotgun just as effectively as an AR. Either one will work just fine for HD work, if you train and stay proficient.
-Jenrick