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.
WWI and WWII did not affect every country in the same way nor equally. For example, in the USA...there was no combat here. Sure the people served in Europe and Asia and elsewhere. But, there was no actual damage done EVERYWHERE on the planet. If THAT were to happen...and the devastation literally covered the entire planet...we'd probably HAVE to ban together to help fix it.
For example...I live right next to NYC, therefore what happened on 9/11 was a bigger deal for me and the people who live in the vicinity. Did it affect the whole country? Yes, but not as much as it affected the people who were IN the building, like my cousin...or me who could look out my window and literally SEE it.
If the Star Trek's version of WWIII is actually devastation to EVERYONE and not just a cluster of people, yeah I think it could change how people think.
And contact with aliens? We would finally realize how freakin' small we are compared to the rest of the universe.