You're ignoring that Star Trek, particularly in its later forms, tells us that a Third World War (which nearly destroyed humanity), and first contact with aliens caused the people of Earth to re-evaluate their goals and they joined together to improve themselves and society.
If the first two world wars didn't do it, what makes you think a third one would? Is it the old saying, "Three time's the charm"? Then I can understand where you are coming from as far as how Star Trek Universe comes into being, but again, reality does paint a different picture.
I somehow believe that in Star Trek Enterprise episode "In the Mirror Darkly, " that would be the mentality of the people of earth at that time when survival, looking out for number one, and having the upper hand in all things like power is how anyone thinks they are going to get ahead.
However, in this reality, Biblical prophesies foretells one third of the earth burning up, and that was just to serve as a catalyst for the NWO and their mark of the beast system. Survival will be all that those people will be thinking about, and yet by adapting that mentality to be united with one cause, Star Trek is still no where in the picture. But there will be a gathering of the armies of the remaining parts of the world to battle the Lord upon His return with the saints at the end of the great tribulation period. ( Which is seven years, starting when they give the command to build the Third Temple in Jerusalem. )
They are already adapting the technology for the mark of the beast system, and that is the biochip to buy and sell in Europe. The biochip is being used elsewhere for medical information like in Mexico and security purposes in companies like GPS tagging. It is not the mark of the beast system yet, until the bio chips becomes the only means to buy & sell in this NWO, thus killing off those that refuse the mark because of the Biblical warning from God that those that take the mark are destined for the lake of fire.
Maybe the Lord will have His people trekking the stars when Satan and death has been done away with forever after they have been cast into the lake of fire. There is nothing in Biblical prophesies about that, but who knows? God only knows.
It's time like these that it wouldn't hurt for fans of Star Trek to do a little research on what direction the world is going and how it is lining up with Biblical prophesies. Maybe they will look as I do in the direction higher than the heavens to Heaven above and the hope of us getting there will not be by a starship, but by the hope in the Son of God, the Lord Jesus Christ as it is His righteousness and His faithfulness that is bringing us there for all those that believe and put their trust in Him.
So... maybe Star Trek may happen when mankind is no longer broken and thus in no need of the Saviour, because they are saved and free to roam the universe for they will never be apart from Him.
Christians can dream after all.
Course, this is in according to my faith on how Star Trek could be, but I just don't see it happening when mankind is not perfect yet to do that trekking through the stars. Murphy's law will more likely come into play than anything else as our fallible nature will hinder us so.