Ambassadors, as direct representatives of heads of government, rank somewhere above cabinet members, governors or five-star admirals. Multiplied by 32, plus the fact that some part of the representatives were said to be in an "explosive" state of antagonism. That doesn't sound like an assignment for an unimportant or inexperienced captain.
He's moving them from point A to point B and making sure they don't kill each other not actually negotiating anything, that sounds like the sort of assignment you give to someone who is available. So it was probably less Kirk's reputation that got him the assignment as he wasn't doing anything too important at the time.
I never really got anything from TOS that the Enterprise had the same importance to the cosmos as the Enterprise-D was in TNG.