I think Cooper was honest and truthful, but some things he got wrong. No one is perfect and in this arena there are no proofs.
The JFK driver probably heard something,M turned around with a pistol to protect the president, saw he had no close threat, and continued driving. As he pulled the gun, light glanced off it, the camera captured that as Oswald's Bullet exploded Kennedy's head.
As for the aliens, the word I got was slightly different. A formation of flying disks went over a gunnery range and the CO ordered the men to shoot them…to no effect. They landed and Ike was playing golf in Palm Springs, so they brought him over. He met with the aliens, but told them we were not ready for contact. They and other species keep examining us mostly at a distance to avoid official contact.
Cooper heard a lot, but it's all from other people, who can get details wrong.
The gun control stuff has been confirmed by events.