The big problem with this part is unknown origin of this Federation. This explanation wasn't given in the games or the anime. There's simply a federation that maintains order throughout the galaxy.

But what I propose to do? To simply give this an origin. If Rick's age is still exists and USA too (Because Mute City is NYC in past) It would be reasonable to assume that Earth in future could face a crisis in which the traditional superpowers cease to exist. Sometime around 2074, the Arab Crisis occurs, during which oil supplies across the globe become scarce, with the only remaining source being the Arab states. Ultimately, there are no resources left for further wars for influence, and states begin to disintegrate, from the United States to China.

Subsequently, following this event, many brilliant people from all over the world founded "Idea." This organization, dedicated to the implementation of new ideas related to technology and medicine, selected the best minds. Some remaining countries were reluctant to let go of their talents, intending to develop what remained of their country, but ultimately agreed to provide resources to "Idea."

By 2100, old programs are revived, including a cryo-intensive care unit, where Rick Wheeler himself is among the patients. Already in the mid-23rd century, a piece of matter from a dead star (which would later become the Power Reactor) falls. Studying it allowed scientists to travel the galaxy and then find similar scientists from another planet, thereby establishing contact. This is where the technology for the metropolis that would become Mute City comes from. Thanks to alien scientists, humanity was able to travel to different planets, discovering new ones and establishing a common language with the local inhabitants. Earth gains control of these new planets. This is how the Galactic Federation was born.

Perhaps I have compressed or cut corners somewhere, but this is certainly better than the lack of information about how this federation appeared by 2450 (according to our version).

Next stage of reworking F-ZERO anime world building. How Galactic Space Federation is founded?