The US government would probably only "open our skies" to airlines whose home countries would grant similar opportunities to US carrier's.
Thanks to asymmetrical bilateral agreements US carriers often have rights of carriage in other countries that foreign carriers either do not have or cannot make practical use of here. If you look at the history of international travel agreements US carriers have benefited greatly from the disproportionate power of their government to set the terms largely in their favor. Even in cases where the agreements would appear to limit the rights of US carriers they are sometimes overridden or simply ignored.
I was not aware of that.....could you give some examples?
The only case that I am aware of, was back when Germany was divided, I believe Pan Am had the exclusive authority to fly between Berlin and Frankfort where they operated the "Internal German Service", or something to that effect. I believe that even Lufthansa was not allowed to fly that route. Not sure of the entire history of that, perhaps something to do with the Berlin Airlift earlier, when the Soviet's blocked the land routes between Berlin and West Germany.......
As for the US having an advantage in some trade agreement, I say that's nice for a change....we usually get the short end of it with countries like China or Japan.....