The fully automatic AA-12 shotgun is only used for military applications AFIK?
As i said, my own real life experience is that you brace shotguns to the shoulder
Nothing your run n gun with 
But in the world of SR5 it would make perfect sense that Pistols, Shotguns and SMGs get a smaller negative dice pool modifier when using them while locked in melee or when you are considered running (or some similar bonus that mostly would affect CQB)
Also,
CoD4 Hardcore Mode. Best FPS evar.
(except maybe CS... back before patch 0.6 or so.... when it was still a free mod to HF)
Actually fired an AA-12.... not a bad piece of hardware... thanks to the unique bolt system, very little recoil so it was very accurate!
The reason why Militaries shy away from shotguns is more of logistic issue and a safety issue.
For the logistics side, militaries quickly found that having dozens of different caliber weapons in the field lead to massive problems in logistics. With so many different weapons and calibers, units in the armies sometimes would not get fresh, usable ammo for their weapons on a consistent basis as their was just too much confusion as to who needed what, where. (cira 1890s to 1910)
By the start of WW1, many armies still fielded a variety of weapons, but he calibers used changed from dozens to just a few. (2 or 3 different pistol ammo, 2 or 3 rifle, 1 or 2 machine gun) But this still lead to supply issues as entire regiments would be sent the wrong ammo! Thus leaving them effective out of the fighting... or "Walking targets" during the march towards the enemy lines...
By the start of WW2, most armies had reduced the number of different rounds down to a small hand full, as well as standardized weapons for all soldiers, instead of this mis-mash of weapons of different sizes, quality and makers....
Today, most modern armies function of just a dozen different calibers in total (and yes, the 12-guage is one of them) thus easing the logistics of supplying ammo.
As for safety..... No other weapon in the world jams more often then a shotgun. Doesn't matter what the feed is on a shotgun, be it pump action, semi automatic or fully automatic, the all have a much higher rate of fouling and failure.
Part of this is the fact that shotgun ammo comes in plastic casings and those casing expand and warp in ways that are not as predictable as brass, thus leading to a fouling of the chamber. the other issue is the gas pressures in shotgun shells is actually a lot less then a standard rifle. This leads to incredibly sophisticated feed mechanics to over come this issue... and even then, shotguns still randomly fail (sometimes lethally for the shooter!).
Range is also an issue, combat shotguns rarely have accuracy past 100m (which is not that big of an issue in CQB)
So, in the field, the shotgun is not a trusted weapon by militaries..... They use them yes... but they would much rather give you an assault rifle. After all if your primary weapon fails.... a soldier is almost next to unarmed!