The stories you posted to try to prove to me that cops kill dogs all the time, half of those were pit bulls, and were probably not owned by little church ladies with edged yards lined with tulips. I'm gonna guess a look at this place with the St Bernard would be enlightening too.
So any dog that comes towards you wagging its tail "looks dangerous" now and deserves to be shot on the spot?
No. You missed the point. You compared the work of a police officer with that of a mailman. If a mailman or parcel delivery so much as sees a loose dog they don't go to the door, they refuse and leave a note in the mailbox. I don't think cops have the same luxury.
And in the story linked the cop was told repeatedly to close the car's door so the dog would not get out. He refused and when the dog got out and approached him wagging its tail, the cop shot it. He had a choice. He chose to shoot first. Same for the top story here, the girl was calling her dog back and instead of simply waiting for it to retreat, they shot it. Now shooting might be justified if you are in hot pursuit, or have a warrant for a violent suspect, but in cases where the cops are exercising their discretion and there are children present, they need to not choose a shoot-first approach.