When should the right to remain silent really begin?

A call is made about shots fired. Police arrive within 1 minute and enter a room. In the room is a dead body which has obviously been shot. A person is standing next to the body with a gun in their hand.

Why should we as a society not have a right to demand an explanation of what happened or what if anything did you do?

Instead the person can invoke right to counsel and remain silent. Not only that but we can never tell the jury that they didn't say anything at the time they were asked.

Does this make sense? Has the 5th and 6th amendment protections gone too far?