^But those other races can't have started out monocultured. Their ancestors must've lived in different regions on different continents and developed into distinct societies with their own languages and customs; it's just that by the time they got into space, their planets had become more unified or more dominated by a single culture.
In which case, maybe the key is that humans achieved warp drive earlier in the process -- or, to put it another way, took longer to consolidate. And thus that kind of cosmopolitanism, as you put it, was a more recent part of our history.
Although I think what ENT showed is that the reason humans were able to bring others together and take a leadership role is because humans were the only ones without a history of enmity with the other races, and thus were the only ones trusted to be neutral and fair.
It could also simply be that younger civilizations are hungrier, more ambitious, more driven to grow, while older, more settled civilizations are more set in their ways or content with their lot. For most of history, China was the most advanced and prosperous civilization on Earth; but it was therefore content to stay as it was, whereas the poorer, more backward English civilization was more driven to expand its influence and increase its prosperity, leading it to rise to global dominance and outpace the older, more settled civilizations of the world.