If there is a "murder", then there is a crime. So you have something to investigate.
Compared to some one in the police state saying something like:
"I do not like tronthetechie, lets spend $10 million dollars on a high powered team of investigators to see if we can find if tronthetechie has ever done anything wrong. We could even kick in the front doors of some of his associates and threaten them if they do not say what we want them to say about tronthetechie"
A just system requires the identification of the crime before an investigation is lauched.
Actually, I feel like you misunderstand the nature of policing in the United States. The officer wouldn't have to go through those many steps to legally kick their way into my house, and then use anything as evidence against me, they just have to cite probable cause.
Probable cause can be ridiculously easy to establish "it smelt like marijuana" "I heard someone call for help" "it sounded like there was an altercation" "The evidence we need for our investigation is reasonably in this location"
I'm sorry to break it to you, but even if there were something wrong the state did while policing, it would be very difficult to prove to the state that the state did anything wrong in the first place.
Shitty, I know, but it is how it is right now.
Americans can't define murder.