Your comment and the OP act like the Tellarites don't like anyone, but that's not really true. The Tellarites and Vulcans got along just fine before Humanity showed up. The Vulcans probably figured out the Tellarites' gimmick and decided that challenging each other's ideas was how they arrived at a logical conclusion. The Andorians were probably too hot headed to get along with people who challenged them on everything, leading to them being in conflict during ENT.

As for how humans fit in, humans may be crazy bastards, but they also occupy a nice middle ground between the others. We can be as emotional as Andorians one day, as logical as Vulcans the next, and a number of our political systems were built on Tellarite-style arguing. Humans can hear out one ally, then explain their position to another in terms they'll understand. We can also tell when Vulcans are secretly being driven by emotion, can leverage Andorian emotions to bring them to a more rational perspective, and debate Tellarites on even ground. Humans are the interpreters and negotiators that got these people to actually talk about their problems with each other and come to a reasonable compromise.

The Federation is created at the end of Enterprise: before that there's a cold war going on between the Andorians and the Vulcans. The Coalition of Planets forms during the show and is dissolved when the Federation is created.

Huh? They didn't hold together a Federation. It didn't exist yet.

Well, the whole point of a diplomatic interplanetary body is to have a common forum of communication to soften their relationships over time, by allowing dialogue and cooperation to occur at some level in a fashion doesn't involve shooting phasers and photon torpedoes at each other. Even if the shooting still happens from time to time.

It's a parallelism with the UN. All sorts of countries who hate each other are part of it and sit at the same table, at some unequal level even. They still shoot each other and commit atrocities towards others and their own people from time to time. But still, the point is that the table exists and is available. It creates opportunities for dialogue and cooperation, that wouldn't exist otherwise, that create opportunities to solve conflict peacefully and soften their differences over time.

There's a ton of Voyager episodes of "a few crew are captured and they're working for this other set of people out here!" But those are generally not a view of another well-developed culture since they're in such foreign territory.

The Cromulents.