I kind of get the idea that the Federation at the time was just this gigantic shitshow.

Within the Federation, humanity has affectively taken over as the prime species, with the other founding members withdrawing a lot to their home systems. Vulcans are barely there in Starfleet and seemed to be judged a lot by their peers for doing so. Tellarites and Andorians are even less prevalent, making a lot of their service a novelty.

So you've got the youngest of the four members leading expansion through their territory and it goes just about as well as you'd think.