The big deal is the dog wasn't attacking. Your attitude that an innocent family pet can be "popped" for nothing, and is no "big deal" if it *is* a pitbull dog, is sick.
Cops shouldn't shoot innocent dogs, even if it is a "pitbull." Your attitude seems to imply as long as the cops are just guilty of shooting innocent dogs that "appear" to be a pitbull, everything is peachy. Rationality is apparently dead, where you live.
I'd say that's inconclusive at best from the video. If you had the gun, and the dog (a pit bull/boxer mix) was running at you barking, what would you do? Negotiating with a dog usually doesn't work very well, especially after it's latched onto your leg.
I don't think it's okay for cops or anyone else to shoot an innocent dog. I do think that this whole thing was handled wrong from start to finish, to put it mildly. The worst thing is the treatment of the family. Until we have cops that are paid a decent wage out there, we will continue to have incidents like this.