ant, I understand your premise, but in my experience it doesn't necessarily work that way. I've noticed a few distinctive things about dogs from when I was a handler. One is, especially during pursuit or active movement, once their "RADAR" picks up on movement, it locks on. Dead on. Most dogs, at least those that I have caught, tend to have a sort of "routine", and you can generally learn when and how a dog will hit during pursuit.
Training them generally to go after an arm, and then making them discern as to which arm they should go for becomes difficult. When the reward is the bite, it is difficult to discourage them from biting the "wrong" arm. You have to be relatively careful in bitework, it is a delicate balance at times between control and aggressiveness. When you are working with natural drive or instincts - their predecessors, when hunting, did not have to deal with weapons - the hunted animal was the weapon.
Sending a dog on a passive person and getting a bite is also a delicate balance, and very difficult. Remember, prey drive functions best in a fight. In this case, the dog will definitely take what they can get. Sometimes engaging in short, noncommital bites to get the movement and fight that they want. I'm not saying it's impossible, but it truthfully is not that encouraged. Several dogs these days are taught "bark and hold" not "bite and hold", especially on passive subjects or those that give up.
Again, some dogs are just naturals. We had a case several years ago where an individual took hostages in a dry cleaning business. After a several hour standoff, he exited the business, holding a rifle. He would not drop the rifle, but was not pointing it at anyone, just carrying it in one hand and stopped. One dog was sent from the other side of the building than the one I was on. The dog perfectly engaged the non firearm arm and drug him to the ground. I did not release my dog, but ran up with him on lead. Mostly because our team followed the dog's apprehension and subdued the suspect immediately. Besides, I didn't want my dog keying on the actions of the big guys in black.
Which brings me to another point. Remember, a K9 reacts to gunfire. Dogs do not discriminate. They may react to the handler's gunfire as well. Even though we train with handler gunfire, if my dog makes a mistake and engages me while in a firefight, I would rather he not be "trained" to engage my gun hand. Since I'm right handed, and the dog deploys generally from my left side, that sort of helps things.
Clearly, these dogs are presented with very hostile scenarios, and their training includes circumventing objects by going through or over them to get to the target. Most dogs, frankly, will try to go around an object for a bite, before they look at going over something. Again, highly trained dogs. Also remember, that judging by their training, they are good at what they do. Dog sends in these scenarios are not the end-all, they are a safer way for officers to approach and deal with armed individuals. Ultimately, the dog's life, although very valuable, is less valuable and more expendable than the officers.
