Hammerhead6814's argument holds no water and is borderline irrational. Let's just do a couple of comparisons of the fundamentals of his argument:
Length:
AR with 16 inch barrel and stock collapsed (and even fully expanded) is a more compact choice than most shotguns.
Shotguns tend to be longer weapons than most rifles unless it is purchased in an exclusive defensive configuration. The average shotgun is actually quite long. Fixed stock means ergos are what they are.
The AK maintains compactness, but the stock is one size fits all, hurting its ergos.
Advantage: AR
Recoil:
AR 15 shoots a low recoiling round and utilizes a buffer tube that largely mitigates recoil. Expect some muzzle rise, but little pressure on the shoulder.
12 Gauge has the highest shoulder recoil available in the home defense department. Not a bad thing unless you have a bad shoulder like the OP. Also, follow up shots may be difficult.
AK has significantly more recoil than the AR, but is not unmanageable. Follow up shots will be slightly more difficult than an AR.
Advantage: AR
Round Power:
AR 15 has the smallest round of the group, but is still a very effective solution. It has been effective on the battlefield.
12 gauge with 00 or #4 is going to be the best stopper you can fire in your house.
AK is a definite step up in power over the .223 and will deliver the goods in a home defense scenario
Advantage: Shotgun
Penetration/Over-penetration:
AR 15 has a variety of options available, from heavy grains to light grains. All are very effective on targets. The lighter rounds will penetrate less, but break up quickly against walls.
12 gauge shot effective enough to stop an attacker over-penetrates on a grand scale. The notion that the shotgun doesn't over-penetrate is a complete myth.
AK is a big penetrator capable of effective depth in an attacker and also going through your house completely.
Advantage: AR 15 provides the best balance
Nighttime/CQB customization:
The AR has the widest variety of CQB options on the market. Red dots, lasers, vertical grips, and easy flashlight attachments make excellent quick CQB deployment possible in any situation
12 gauge shotgun can be outfitted with lights and possibly a red dot. Red dot sights must be able to withstand recoil of the round
AK's offer the most difficult path to upgrade to a good night CQB. Awkward scope mounting. No rails in traditional configs for adding lights or vertical grips. It can be done, but it isn't done often, and for good reason.
Advantage: AR by a huge margin.
Assuming reliability is good on the weapon selected, ammo prices are almost a wash. Of these three options, I think it is pretty clear that the AR is the obvious choice. Of course, there are many more options than just these three selections. If not an AR or shotgun, I would most likely select a pistol caliber carbine.
Length:
AR with 16 inch barrel and stock collapsed (and even fully expanded) is a more compact choice than most shotguns.
Shotguns tend to be longer weapons than most rifles unless it is purchased in an exclusive defensive configuration. The average shotgun is actually quite long. Fixed stock means ergos are what they are.
The AK maintains compactness, but the stock is one size fits all, hurting its ergos.
Advantage: AR
Recoil:
AR 15 shoots a low recoiling round and utilizes a buffer tube that largely mitigates recoil. Expect some muzzle rise, but little pressure on the shoulder.
12 Gauge has the highest shoulder recoil available in the home defense department. Not a bad thing unless you have a bad shoulder like the OP. Also, follow up shots may be difficult.
AK has significantly more recoil than the AR, but is not unmanageable. Follow up shots will be slightly more difficult than an AR.
Advantage: AR
Round Power:
AR 15 has the smallest round of the group, but is still a very effective solution. It has been effective on the battlefield.
12 gauge with 00 or #4 is going to be the best stopper you can fire in your house.
AK is a definite step up in power over the .223 and will deliver the goods in a home defense scenario
Advantage: Shotgun
Penetration/Over-penetration:
AR 15 has a variety of options available, from heavy grains to light grains. All are very effective on targets. The lighter rounds will penetrate less, but break up quickly against walls.
12 gauge shot effective enough to stop an attacker over-penetrates on a grand scale. The notion that the shotgun doesn't over-penetrate is a complete myth.
AK is a big penetrator capable of effective depth in an attacker and also going through your house completely.
Advantage: AR 15 provides the best balance
Nighttime/CQB customization:
The AR has the widest variety of CQB options on the market. Red dots, lasers, vertical grips, and easy flashlight attachments make excellent quick CQB deployment possible in any situation
12 gauge shotgun can be outfitted with lights and possibly a red dot. Red dot sights must be able to withstand recoil of the round
AK's offer the most difficult path to upgrade to a good night CQB. Awkward scope mounting. No rails in traditional configs for adding lights or vertical grips. It can be done, but it isn't done often, and for good reason.
Advantage: AR by a huge margin.
Assuming reliability is good on the weapon selected, ammo prices are almost a wash. Of these three options, I think it is pretty clear that the AR is the obvious choice. Of course, there are many more options than just these three selections. If not an AR or shotgun, I would most likely select a pistol caliber carbine.