Yeah the government is capable of lying and telling the truth as well as the secret third thing (just being wrong unintentionally). They can even sincerely believe something that they have no good reason to. In short it's an organization made of people with policies made by people.

The government was wrong about 9/11, and they lied to protect their ally Saudi Arabia. But it wasn't an inside job, the evidence just doesn't lean that way.

JFK is scary yeah, but a lot of US presidents have been shot at. 4 have been killed in the process of being shot at. Of those 4, none had the support of organizations that they aligned with. McKinley's assassin famously was asked not to do it by Emma Goldman. But then you zoom out and suddenly you see that none of the significant attempted presidential assassins had backing either. Most of them, regardless of success, were isolated, radicalized, and mentally unwell. Enter a marine marksman who hopped ideologies notably often who had been disillusioned in his time in the USSR hoping to get in good with Cuba, but not willing to believe them when they said not to kill the president. That he killed the president is more believable to me than that the CIA killed a president who gave them a blank check and no restrictions, or that the mob he was affiliated with decided to risk that heat, or that any other number of organizations decided to assassinate exactly one US president and chose the wildly popular JFK with his dirty, no nonsense, effective VP LBJ as the one to be assassinated. If any of them had killed Kennedy, Johnson would've had both their heads.