I think it’s more about JFK’s legacy and his extra-marital affairs.
He got VD and his family don’t want his medical history to be revealed.
The secret service in the follow car turned toward the sound of the second shot and accidentally fired his ar-15 fired hitting JFK. The time between the second and third shot could never be duplicated. Doctors reported bullet fragments in JFK brain and brain disappeared. Every person in the cars following smelled cordite from a fired gun.
They never ever want the public to know that the SS shot JFK.