TDC, I doubt that the difference in pressure from shotgun ammo out of the same box would result in a stoppage. If the ammo will cycle through a pump without issue, it will cycle through the semi-auto. If the semi-auto gets tied up due to an ammunition issue, that same ammo will tie up the pump, and time will be spent clearing the jam. When I chronographed 12 gauge slugs I was actually pleasantly surprised how consistent some of it was. I've never seen properly stored factory 12 gauge ammunition that was so inconsistent that it would effect the cycling reliability of an inertia operated semi-auto shotgun, or a gas gun for that matter. Short stroking a pump gun in a fight could cost you your life, and the semi-auto will not short stroke.
As to the ability to carry on a fight with a gunshot wound that has broken your arm when you are armed with a pump shotgun, there is no training in the world that can improve that situation. Once discharged, the gun must be held vertically with the butt on the ground, the slide moved rearward to eject the spent cartridge, #### the action, and bring a round from the magazine tube onto the elevator. Then the slide must be moved forward to raise the elevator, chamber the round, and lock-up the bolt. It must be remembered that these are rack guns, and might not be as slicked up as a personal firearm might be. All the while you are trying real hard not to get shot again, you are trying to bring enough fire on the target to protect your buddies, and hopefully stop the guy who is trying to kill you with rapid rifle fire.
The FBI is probably one of the most gun oriented LEO operating in the world. I doubt that the FBI's training is lacking, but training does evolve, and this particular scenario has been given a great deal of attention. There was some discussion about dropping the shotgun from service altogether in favor of AR-15's, but I don't know if that was followed through or not. Given the extent of his wound, the fact that Mireles tried to fire at all says much for his character, but clearly had his shotgun been a semi-auto, the dynamics of the fight would have been changed dramatically.