It has a gun because pilots don't feel comfortable without a gun. It makes no sense, but fifty years ago the USAF jumped the gun a little bit on air-to-air missiles, and whenever someone points out the gun is a stupid thing to include on a modern strike fighter people bring up this fifty year old problem.

The A version of the F-35 has three seconds worth of gun ammo. It would only be two seconds, but there's some lag between the trigger pull and the gun being fully spun up.