30 rnd Shotgun mag 480 pellets downstream, each 40.5grains, roughly 2.75 pounds of lead.
30 rrd SMG mag, 30 . 357 bullets downstream, each 115 grains, roughly half a pound of lead
No, they don't match magazine to magazine, at all. you need five 9mm mags to match the weight of shot
That's kind of my point, you still have to carry all that shot around

! Also since the number of actual rounds fired is the same unless you can show me that each 12g round is 5x as effective in doing its job as a .357 round (you compare the .357 weight but then talk about 9mm, not that they're that different in this). Just comparing weight of stuff going down range isn't a very good way of looking at this. Or put it this way do you think in abstract the the 7.62 nato is roughly 3x better at doing it's job than a 5.56, what about a musket ball?
This leaves aside the weight of the projectile isn't the whole story you have the rest of the cartridge, you have the magazine or drum they're in, the weight of the gun firing this stuff. Again seriously have you handled a 30 round 12g drum, even empty it's an awkward shape. The 32rnd drum on the AA-12 weighs 2.1kg, how many of them do you think you're going to carry in comparison to 9mm or even 5.56mm magazines when you include it in with the rest of a load out. The AA-12 itself's empty weight is almost double that of the M4.
Again shotgun with shot shells are severely limited in the roles they can operate in. So you have a very specialised weapon compared to short AR. So you say ok we'll have rifled slugs as well. Well OK that's now 2x ammo types each with all the issues above and the issues of carrying 2x different round types. Or you go with something fancy and expensive like fragmenting explosive shells, but that runs into supply and resource issues, and how much of that rounds utility is just making up for a shotgun shell's deficiencies i.e. running to stay still?
Basically people have been trying to shoe horn shotguns into a significant combat roll for well centuries actually, and it's never really happened other than very specific and limited examples. The reason is that while there might be some specific situations in some conflicts* where having one has had benefits in the context of the time, the down sides of getting it there and actually operating with it in combat out weigh it.
Atchisson's being trying for decades and it's not happening. You might consider it a lack of imagination by overly conservatively thinking military assessment boards (and well that's by no means an insane idea in general), but I think it's more that the inherent issues of shotguns hasn't actually changed in that time.
*skirmishing riders in the send half iof C19th with shortened double bore, trench raiders in WW1, spec forces in Vietnam jungles or other CQC (spec forces use all sorts of weird and wonderful and mission specific stuff with way more resource leeway than most though, they also are more able to withstand heavy load outs and adjust them as they see fit)