As the law is written, police officers in the US are not allowed to shoot or kill people in circumstances where non-police cannot.
This is why you hear the common police trope of “I feared for my life”.
In practice, of course, the prosecutors will aggressively prosecute people defending their own homes from intruders, and fail to prosecute police engaging in premeditated murder; but that of course is a different story. The legal framework, in theory, provides them no special privileges to kill.
The police, by law, are allowed to create circumstances that non-police are not.
However, even if they weren't, it's irrelevant to my point, that people, including but not limited to the police, have powers that come from their membership in an abstract entity, and it's perfectly natural for them to have regulation of those privileges as well.
This is why you hear the common police trope of “I feared for my life”.
In practice, of course, the prosecutors will aggressively prosecute people defending their own homes from intruders, and fail to prosecute police engaging in premeditated murder; but that of course is a different story. The legal framework, in theory, provides them no special privileges to kill.