I noticed a valuable error in your reasoning. I am a fair man, so I will put this down to ignorance, but not an intentional one.
You say why haven't they showed themselves to the elite>
The military have I would suspect more scientific members working for them than what we have come to realize. What exactly do you constitute as ''elite members''?
If the military spot them on radar, how more elite do you need to be? Secondly, Hawking does not have a valid point - unless that point is validly narrow.
The military is made up of a LOT of members. As is NASA, the government, etc.
Wanna know how secrets get kept? Simple- Most any secret is so mundane to the average person (Who probably wouldn't understand why the shape of a submarine propeller was such a big secret anyway, for example) that it's not worth telling anyone.
But a juicy secret like Alien Visitors? Out of ALL those people? Could YOU keep your mouth shut about that?
Radar and UFO's spotted by the military is something that's been exhaustively researched. Not just by UFOlogists, sensational Discovery channel or Visitor WannaBeleivers, mind you, but by many agencies on the account of National Security... About Other Nations.
Not aliens.
Their concern was simply Other Nations. They Really wanted to Know if Russians were doing fly by's.
So you can imagine that they were more than a little inquisitive.
As it stands, Alien Visitors has about as much support as Ghosts.
Sure, sensational media will hype it up as fluffy as possible. Lots of people out there believe in Ghosts, too.
But there is no hint, no indication, no sign whatsoever that alien visitors have dropped by.
Is it possible that they have? Sure. But not very likely.
Hawkings point is very valid.