What Doesn't the President Know?
Do you think there's classified / top secret information that the POTUS doesn't know? Like, is the CIA or some other agency keeping UFO information from the president or some crazy shit? Does the president know what really went down with Kennedy? Or is everything declassified once the new president is elected? What do you think?
I think there is stuff the president doesn't know, I'm not sure what, but it would actually make a bit of sense if you think about it. Not all high ranking positions in the government have a set term, right? The president does, the most he'll be in the White House is 8 years, and it's possible he might only be there for 4.
Thus the job is a bit of a revolving door and on classified matters where the least amount of people that know about it the better..they might not want to make people aware of certain information if they might only be there a few years and then you have yet another person you need to bring into the loop once that person leaves.
So for instance since you mentioned JFK, I think if there was some kind of government conspiracy to kill the president then it is most likely they wouldn't inform a different president of that fact. It would basically be like going "yeah so we will shoot you in the face if you displease us".
I think there's stuff that'll be brought to the President's attention if they need to make a call, but just quietly runs in the background otherwise.
If you're president, you do get given all kinds of high-level information, so it's not a trust thing due to the term limits. After all, a president has a gigantic amount of information the US doesn't want out regardless.