This is a nice dream. I'm cynical enough to fear that our species is simply incapable of ever presenting such a united front for long, but I hope I'm wrong. I've always believed that uniting under a single planetary government is essential if we're going to expand out into space and survive in the ultra long-term. I like to think of myself as a citizen of Earth first, and United States second.
The EU was a bold step in the right direction, but Brexit has dealt it a terrible blow from which it may not recover. Adopting a single agreed upon flag for space missions is an excellent idea, and I like the symbolism of the design you've highlighted. Now if only you can get NASA and the other major space agencies of the world to pay attention to this!
I am glad you share my hopes for humanity. I believe that one day, we will be a planetary civilisation, not today, not tomorrow, but some day. Maybe 1000 years in the future, but I think that is a natural step in any intelligent civilisation's evolution.
About NASA taking notice....Actually I read that NASA helped with this project along with some other companies like LG. So, the idea is out there, waiting to get adopted! :)