3 MOA is on the upper end for an AK, but still not amazing. I'm just saying when you compare it to, say, an AR performing at 1-2 minute of angle, I don't see why there's such an attraction to the reputed reliability of the AK when you're sacrificing any ability to make a truly long distance shot.
The best point of comparison for a 7.62x39mm AK would be .30-30 class rifles, such as the quintessential iron-sighted lever-action .30-30, which is comparable in terms of ballistics and accuracy. For a civilian .30-30 class rifle, there are very few instances in which you'd practically need to shoot at more than 200 or 250 yards. 7.62x39mm is not powerful enough to hunt deer with beyond 125 yards or so (if you use 1000 ft-lb as the energy cutoff for a humane kill of a deer-sized animal), and defensive use is almost always inside 100 yards and usually inside 25. If you truly need long-range shots, then you'd want to begin with a cartridge that has a flatter trajectory (like 5.45x39mm or 5.56x45mm/.223 Remington), and if you need power at range then you need to step up quite a bit in caliber anyway, which is why .308 and up rule that domain.
You also have to separate the caliber issue from the merits of the platform itself. The best comparison to the M16 is the AK-74M (5.45x39mm) or AK-101 (5.56x45mm NATO), rather than 7.62x39mm. By all accounts, the 5.45 and 5.56 AK's are much more accurate than the 7.62mm models, and they definitely shoot much flatter.
As far as the "why" question, the attraction of the AK is that it is a very economically priced, very reliable carbine that shoots very economical ammunition, yet is versatile enough to use for plinking, USPSA/3-gun type competition, home defense, and light hunting. It is also mechanically quite simple and is exceedingly durable.
I do like AR's, and if I had to choose one or the other I'd probably choose an AR, but the AK is a very, very good rifle when compared to a Winchester 94 or a Ruger Mini Thirty, especially when cost is a factor. The AR typically comes in at a much higher price point.
Also, bear in mind that I've heard that most of the civilian AK knockoffs are substantially more accurate than the "real" licensed production models, which makes sense.
I doubt this is the case, only because a lot of civilian AK's (including mine) use milspec barrels, and until very recently many were built with actual used military barrels that came in as parts kits. So accuracy with the same ammunition should be the same. It is possible that the lack of a slant brake on many U.S. civilian models may cause a slight accuracy improvement, though.
I think the reputation for AK in accuracy may stem from the fact that U.S. forces have commonly faced (and easily beaten) poorly trained insurgent forces using either completely worn out AK's or bootleg copies made in a cave somewhere, and those weapons are often wielded by untrained individuals whose manual of arms seems to be "set selector to Auto, hold gun well below the line of sight, yank trigger, and pray for hits." If 30-year-old M16's with the rifling shot out were issued to people who think "firing from the hip" is an effective way to wield a firearm, the M16 would be equally inaccurate. I suspect that Third World ammunition may also be widely varying in quality.
In the hands of capable shooters (whether U.S. civilians in the case of non-automatic civilian market guns, or first-tier Russian military in the case of the real thing), and used with decent ammunition, the AK is quite a capable platform, IMO.
Here's my civvy AK, a basic Romanian SAR-1. It's a 2002 model (ban era), so it has a smooth muzzle instead of a brake, and no bayonet lug.