Yes, police have far greater immunity.
They are allowed a few days to rest before being questioned about events, because police know about PTSD and how the brain may have events out of order immediately after a traumatic event. (That’s why you never give any statements about the details of what happened until later, through your lawyer, a few days later, your memory will be better, and if you incorrectly recount something but later change your story because your memory gets it right, you look like you’re changing your story and hiding stuff).
Officers are officers of the court. They are professional witnesses. They are presumed to be more truthful than you. Officers have other officers that will lie for them. They work with these people and count on them to watch their backs and would rather protect a fellow officer at your expense rather than being ostracized by the rest of the department. They often carry ‘ham sandwiches’ - throwaway guns and knives to plant on people if needed.
They will generally back who they work with over a citizen they know nothing about. I’ve had family on the police force and know what I’m talking about.
In addition to the practical benefits you described, they also have ,b?qualified immunity, which nobody in the general public has. This item, qualified immunity, shifts the balance point for justified use of deadly force. Given the same events and same witness testimony, a person with qualified immunity is much less likely to be convicted. They must be acting egregiously wrongly, basically killing "in cold blood" for no good reason. All mistakes are excused.