Actually, it was kind of the opposite. Most Assassins and attempted assassins are mentally ill. Let's start with your list.
James Thomas Hodgkinson - Had a long history of mental illness, including firing his gun in the neighborhood. He had been living in his van for the six weeks before the shooting.
Jared Loughner was mentally ill and didn't have a political affiliation.
The guy who never got anywhere near Kavanaugh turned himself in because he knew he was mentally ill.
Rand Paul's neighbor just didn't like him.
Okay, so let's look at other attempted assassins.
John Hinkley was nuts.
Arthur Bremer - Declared mentally ill before he tried to shoot George Wallace.
John Schrank, who attempted to kill Teddy Roosevelt, said he was visited by the Ghost of William McKinley.
Giteau, the guy who shot Garfield, was an absolute nutter.