In the History Channel JFK assassination series, an individual who claims to have been present at the Walter Reed autopsy of Kennedy describes a number of very strange things that happened during the conduct of that procedure.
Among other things, he said that there were non-medical personnel in the room as observers, and at least one of these individuals had veto power over every step of the autopsy.
Given this involved an assassination of a President what you described makes perfect sense to me. Trust no one and place observers with the President’s body wherever it goes.