For one thing, all the conspiracy theories surrounding Kennedy’s death would have about 10x as much credibility than they do in our reality. The assassination itself can be waved away as a lone-wolf tragedy IOTL, but if it had happened on the exact same day as a plane crash that wiped out all of Kennedy’s cabinet, I’d be surprised if anyone—the US government included—wouldn’t have immediately interpreted the events as some kind of orchestrated attack on the US’s line of succession.
Under this hypothetical, this is what the presidential line of succession would have looked like:
Speaker of the House: John McCormack
President Pro Tempore: Carl Hayden
Secretary of State: Deceased
Secretary of the Treasury: Deceased
Secretary of Defense: Robert McNamara
Attorney General: Robert Kennedy
Postmaster General: John Gronouski
Secretary of the Interior: Deceased
Secretary of Agriculture: Deceased
Secretary of Commerce: Deceased
Secretary of Labor: Deceased
Presumably, the surviving Cabinet members along with the Speaker of the House and President Pro Tempore would have been immediately taken into protective custody if that plane had crashed. In reality, it turned around and returned to the Honolulu Airport from which it had just left after the passengers were notified about the events in Dallas.