Remember, we saw a number of episodes where Starfleet sent Picard, specifically, to achieve some difficult diplomatic feat, like the negotiations he completed at the start of "Captain's Holiday." And we're also talking about the only human to become a Klingon Arbiter of Succession -- that's one hell of a diplomatic feat, in terms of the reputation and fame it would give him if nothing more.
Sure, you can argue that in principle there should be more prominent diplomats, but hypotheticals don't trump what's in the show. TNG came to be a show that was very much about politics and diplomacy, and it wasn't about Ambassador So-and-So, it was about Jean-Luc Picard and his crew. So by the nature of series storytelling, Picard became the most important diplomat around. Yes, maybe it's not the most realistic scenario, but it's the fact on the ground, and the best we can do is rationalize it.