There are things the military has that even the President is kept

Yes.

There are some Presidents that I would not trust even with "common" knowledge. To think that some of them might have been told about Secretive "Alien" Information, scares the poop out of me.

According to the Urban Legend, the Philadelphia Experiment resulted in some very unpredictable, scary, and tragic "side effects". The last thing we need is some idjit in the White House, being told this stuff really happened, and then deciding to start the experiments again, and ends up destroying our planet, by creating a Black Hole.

If such things really did happen, and were covered up... Leave it buried in the X-Files until "Alien spaceships land of the White House Lawn".

That is the only time when most people need to be told.

I don't know if there is proof of Alien life, but I do understand why it would be continued to be covered up. I am sure there are real X-Files departments in many countries all over the Earth, hiding knowledge from our society.
Most of the people in the world is not ready for such revelations. There would panic, and there would be total Chaos... Especially when our Government would be forced to say things like, "There is nothing we can do to protect ourselves from them... They are going to keep coming, and they could wipe us all out at will."