Today the police lie all the time. I doubt very much that they identified themselves clearly as police - except, perhaps, after they shot him.
If the target was a few doors down the dog-shooting doughnut-eaters could have knocked on the victim’s door to let him know what was happening.
Even if they did, so what? It's not like that idea would never occur to home invaders, and why should it be the homeowner who pays the cops' salaries to take the risk.
And then what about this factor:
“The poorest man may in his cottage, bid defiance to all the forces of the Crown. It may be frail, its roof may shake; the wind may blow through it; the storm may enter; the rain may enter; but the King of England may not enter; all his force dares not cross the threshold of the ruined tenement.” - Pitt the Elder, speaking in the House of Lords, 1763