I'm thinking you answered your own question.
J Edgar is said to have made a statement that the American public had to be made to believe that Oswald acted alone.
He acted alone. He was helped down the road to action by the Mafia, IMHO. It was still his call, and 3 shots from a Carcano. Not Hoover, not LBJ, not space aliens, no French (seriously?) team, a single rifleman with no one else in the room making three easy short range shots.
If the govt came out and blamed the Mafia, how many Italian Americans would have died and or been burnt out of their homes and businesses in the ensuing months? I will bet that those who had any involvement were eliminated in pretty short order wherever in the world they were by those who work off the books.