Consider what happened in Cuba in the early 60's. That could have easily set of a nuclear war, destroying most of the planet. The Bay of Pigs was an absolute failure on the US's part, seriously damaging Kennedy's reputation as President so early on in his administration and allowing for the USSR to put weapons so close to the actual US. Any government in their right mind would use the Wraith to secure that operation was a complete success no matter what.
And then again with the Cuban Missile crisis. Could potentially set off a nuclear war, and stopping a nuclear war be anyone's number 1 priority. The US would use the Wraith immediately and not even bother with co-operation with Khrushchev.
That's just one moment in history, but there are plenty of other examples. There's no way the US would allow Bin Laden to live for 10 years after 911, so I doubt any secret operations would take priority.
How would the other countries compete? Wraith's existence is a fluke; he came from the freaking sky. You can't create another one of him no matter how hard you try. Keeping him secret is done for no reason other than the plot demanding it.Wraith is being kept a secret so he can be used for whatever purpose the U.S. wants to use him without causing an international metahuman arms race.