I don't know why that was not a consideration when they were trying to figure out who killed him. If you take into consideration that the killer had to be someone who knew how and had the strength to snap his neck, that narrows the list considerably.
And why did the detective allow the body to be removed? He knew that the guy was dead and he had a scapegoat, why wouldn't he insist on exposing the murder (unless he was worried that Nate had seen him, but they didn't throw that bit of exposition in there)