Complaint about a 1953 Philip K Dick story - Imposter

With spoilers, of course:

OK, so it's in this robot anthology: Souls in Metal, published in 1978. It was in the bookcase at home. It reprinted robot stories written between 1938 to 1974.

Imposter, by Philip K Dick, is a very dramatic story. The narrative is in the third person, but the point of view is limited to one character--a man, or so we think, who's working on an important military project. His name in Olham. He seems like just a normal guy, with a wife, a love of the outdoors, and a longing to take a break, a camping trip, in the woods outside of town.

But, no, we find out he's suspected of being a robot--a spy robot--sent by the alien enemy. The robot has landed in a spaceship in the woods outside of town, and murdered the main character after first copying out the man's memories and impersonating him. The robot contains a bomb, that will be set off the moment the man utters a triggering phrase. The danger is that the robot will carry the bomb into the heart of the Secret Military Project and set it off.

The main character knows they must be wrong, and does his best to prove who he really is. After clever plot twists, he escapes and makes his way to the spaceship where the robot landed. He wants to prove the robot failed his mission. Instead, he finds his own dead body. In despair, he says, "But if that's Olham, then I must be---"

Oh no! That was the triggering phase. The story ends with:

The blast was visible all the way to Alpha Centauri.

And I thought, "Wow," and then, "Huh?"

Because the triggering phrase was obviously the result of seeing his own dead body, that is, he was set up to explode exactly where the robot found him in the first place.

Which made me wonder, why would the aliens bother with the whole robot thing, if the blast from exactly where their spaceship landed would be seen all the way to Alpha Centauri? All they needed was to set off the bomb immediately. What happened to the purpose of the robot to take the bomb into the heart of the Project? Did the aliens then fail? That wasn't the implication. The implication was that blast blew up all of Earth, kabloom. So, anyway, it doesn't make sense. Did I miss something? If not, then,

How did Dick get away with it?