Keep this in mind too: Star Trek, while originally attempting to portray humanity in the future, inevitably created canon. Thus, there is a world war 3 in the star trek universe that occurs around 1999 (I can't remember the exact time frame). So It makes sense that 20 years after the world "ended", people would be pretty scared and do anything to try to right civilization.
In the end, as the original post implies, Star Trek is no longer a vision of the future, but rather a sandbox to put humanity into and see how they react.
The Eugenics Wars ran from 1996-1999.
World War III was sometime in the mid-2000's with First Contact with Vulcans occurring in 2063.
Personally I don't remember Khan being launched into space, but it is kinda funny when you look back and think that they sort of thought that was going to be possible in the late '60s.
In the end, as the original post implies, Star Trek is no longer a vision of the future, but rather a sandbox to put humanity into and see how they react.