I also think OP underestimates the value of personal ties among the leadership. Anduin has strong ties to just about every one of the Alliance's leadership, excepting Alleria, Turalyon, and Tyrande. His relationship with Alleria and Turalyon is one of respect, and that's why Turalyon was the one appointed regent (I know he's technically king of Stormwind but until such a time Anduin officially abdicates or Turalyon is officially coronated in a written source, his role is more that of a regent to the throne). Interpersonal ties make diplomacy significantly easier, as seen with Anduin's ability to steer Moira and Genn around to his way of thinking versus how badly he failed to bring Tyrande around, enough so that he had to delay his plans to provide Tyrande with suppport when she went off on her own.
Nobody else in the Alliance really has as many personal links among the leadership in the wake of Varian's death but Anduin. And with him taking on older, wiser minds with military experience as advisors, and (unlike Garrosh) actually listening to them, he's able to make up for his inexperience, allowing him to fill a more symbolic role as High King while also pulling his weight in the duties associated.