Getting classy drunk while on the battlefields.
TL,DR: Alliance established the position of the High King and tied it to the throne of Stormwind to be able to capitalize on its military (which is the largest and the strongest within the Alliance) for rapid mobilization, well maintenance and political stability in leadership. Anduin himself is currently unchallenged and accepted because he has good publicity and managed to build close and respectful relationships with the rest of the Alliance leaders before his ascension.One of the often raised complaints or concerns regarding Alliance is why a 17 year old human boy leads the Alliance instead of any one of the more experiences leaders. The answer lays with the structure of the Alliance as an institution.When Alliance was first formed, it was exactly that: A military alliance of independent kingdoms and nations. The organization was structured as such that a Supreme Commander was at the head of its military forces, and its military forces were consisted of whatever troops each individual sovereign nation saw fit to send. Supreme Commander had the power to call Alliance nations into action, but no power to force them. If they sent armies, SC had armies, if they didn't send armies, SC had nothing. Back then Anduin Lothar was made SC because he was seen as a respectable, able, experienced and a semi neutral figure among 7 human kingdoms. After Anduin Lothasr's death, the mantle passed onto Turalyon who fulfilled pretty much all of the previous requirements. However, after he got stranded beyond the Dark Portal, the mantle fell onto Therenas Menethil II.During Therenas' tenure, it was rather expected that Arthas would bear the mantle after him, simply because he was Therenas' heir, but no rules or arrangements existed at the time to make it official. Events in W3 made all of those assumptions null, as Lordaeron fell to the scourge with Arthas as the one bringing it.During the events of W3, Alliance had basically collapsed and disbanded in all but name. At some point there was basically 3 "Alliances" at the same time, some of them maintaining a minimal amount of contact, mainly the New Alliance of Garithos and the Alliance in Stormwind, while some straight up fought one another, mainly the Alliance of Theramore and Alliance forces of Kul'Tiras, which was allied with Stormwind at the time. After the death of Garithos and his forces as well as Daelin and his forces, Stormwind and Theramore were left as the successors of the Alliance of Lordaeron. Theramore and Stormwind did manage to come to terms and establish a unified Alliance. During this time, the rebuilding was mainly done by two people: Jaina Proudmoore and Bolvar Fordragon. Jaina was Lady of Theramore, a sovereign ruler in her own right while Bolvar was the Regent of Stormwind, basically the defacto ruler of another independent kingdom. As a result, the old Alliance structure was mostly preserved, mainly the fact that the scope of SC's powers and authority was unchanged.There was, however, a change that Alliance did not get to utilize during the Third War: The SC was also the ruler of one of the Alliance nations. As a result when SC wanted an army, they actually got one, the Stormwind Army, and they had it with all of its might. With Stormwind being the largest and the martially strongest of the Alliance nations, this condition proved to be a huge boon, as Alliance was able rapidly response to scourge and deploy efficiently in Northrend.During the events of Cataclysm, several weaknesses in Alliance structure was exposed, not all of them yet addressed, but one saving grace was once again the fact that armies of Stormwind was ready to fight even if everyone else were not. This solved several problems the old Alliance faced:They no longer needed to wait for everyone's forces to come and assemble. They get to make quick and organized responses.They didn't need to rely on majority nations to consider the threat agreeable to mobilize, just Stormwind was enough to make a counter for seemingly small incursions, crushing them before they become bigger threats.Stormwind was already a large, highly martial and quite a wealthy kingdom. They made sure they were safe by always upkeeping their armies. This meant that Alliance always had a massive, well equipped, well trained, well managed fighting force, united and ready to fight at all times.As a result, the idea occurred to make this a permanent feature. There was another problem that Alliance also saw as crucial to solve: Stability in leadership. Third War was basically a nightmare, the lack of a clearly line of succession meant that when Lordaeron went down, Alliance was left without direction. Basically suffering from headless chicken syndrome, because there was no rule and no rhyme to how to chose a leader, anyone who grabbed a piece of influence declared themselves as the new big guy, with the likes of Garithos getting to rule. A system was needed, a reliable one that could be rapidly applied, without losing the benefits of having the ruler of a nation as the SC and without getting Alliance leaders at each other's throat over the position. So, the position of the High King was established: Having a king/queen made sense to Alliance nations as so many of them were ruled by monarchs or other similar autocrats. This title was tied to the throne of Stormwind itself, making the King of Stormwind also the High King of the Alliance. The powers of the High King remained unchanged from that of SC. As a result:Alliance got to keep all the benefits I mentioned above: Ability to rapidly mobilize, maintain a large fighting force at the ready, respond to threats without a need for large consensus.Alliance nations kept their independence and individual rulers of those nations kept their sovereignty.The leadership of the Alliance settled in stone, in a manner that allowed rapid succession without dissent or disarray.The crown of Stormwind gained all of the Alliances power and support to keep it stabilized.The stability doesn't bring stagnation, in the eyes of longer lived races at least. Humans' short lives make their reigns flexible. If they don't like the current High King, oh well he will die in 6-7 decades at most, that's nothing to several millennium old people.All of these bring us to the current situation: High King Anduin Wrynn. At age of 17, he is old enough rule Stormwind without a regent and thus old enough to lead the Alliance without a regent too. He ascended to his position due to the new structure established and finalized during the events of Mists of Pandaria. All of this, however, may not explain why several millennium old leaders of other races are okay with him personally. The answer to that lays with Anduin himself:He has established relationships ranging from respectful to downright lovingly doting with almost all of the Alliance leadership. Jaina Proudmoore is his "auntie", he called Magni Bronzebeard "uncle", he was Velen's pupil and seen by him as a sort of potential savior, he personally stood up for Moira's life and right to rule, has close relationship with August Celestials, is basically seen as a surrogate son by Genn Greymane. The only people he has no overtly familiar relationships with are Tyrande, Malfurion and Gelbin Mekkatorque. And even with them he has familiarity as he interacted with them on several occasions while accompanying his father.He is a powerful wielder of Light, which endears him in the eyes of majority of Alliance population as they are followers of Light in one shape or form. He is exceptionally accepting and tolerant which makes him liked by those not really fitting with the common cloth of the organization too. He proved himself, on more than one occasion, to be a brave and kind person with strong morals and virtues that Alliance as a whole tries to uphold.The rest of the leaders, so far, have no reason to dislike him. For Velen and Jaina he is their protege that they help shaped into to the person he is now. For Genn, he is the son of the man he followed and admired that he feels responsible for and proud of. For Moira he is the guy who saved is life and made her current power possible. For the rest of the Council of Three Hammers, he is the guy who gave them a chance to unite their long divided people. For Aysa he is the son of a man she respected greatly and a personal pupil to August Celestials. For Tyrande and Malfurion he is the son of Goldrin's chosen, with strong divine powers in his possession and a great respect for all living things.All of these factors combined explain why Anduin is High King and why that ist not a point of contention within the Alliance, yet. We don't know what future will bring: Anduin's peace loving persona might drive scorn and perception of incompetence in the time of a global war, or his shift to a more militaristic approach may alienate his old moderate allies. Or he may come out of all this a hero and a good leader that many within the Alliance hopes he becomes. Only time will tell.TL,DR: The Alliance established the position of the High King and tied it to the throne of Stormwind to be able to capitalize on its military (which is the largest and the strongest within the Alliance) for rapid mobilization, well maintenance and political stability in leadership. Anduin himself is currently unchallenged and accepted because he has good publicity and managed to build close and respectful relationships with the rest of the Alliance leaders before his ascension.