Exactly. Massad Ayoob says in one of his videos that, in addition to not talking to the police without a lawyer after you've shot someone in self defense, you should tell the police that you want to go to the hospital. I can't remember exactly, but I think the police are not allowed to try to question you in the hospital. Also, this means that you won't be brought directly to jail, I think. I need to watch the video again to get all of this exact, but he was pretty emphatic that telling the cops you need to go to the hospital because you feel your heart racing, or something like that, is a wise move.
I do know an officer involved in a shooting is given that amount of time prior to making an official statement. They’re also provided a Union rep and an opportunity to review all the body cam video.