Jarl is approximately equivalent of Duke, not King.
It's really neither here nor there but the term Jarl and King are used interchangeably in lore documents referring to the Jarls of Skyrim. What really matters is that in the absence of a lineal heir to the throne of the High King of Skyrim it's the Moot that elects the next High King. Without the support of the Moot there is no legitimacy to Ulfric's claim and he would be seen by many if not most Nords as a usurper.
Bear in mind that the Civil War is something Ulfric claims he is waging for the goals of the ejection of the both the Thalmor and the Empire from Skyrim to reestablish the sovereignty necessary to restore open worship of Talos in the land. Although there is ample evidence to draw the inference that another goal is to become the High King of Skyrim, Ulfric never overtly and explicitly states that as a motive and he recognizes the importance of respecting the authority of the Moot in legitimizing his rule enough to frustrate its meeting with the Civil War and to defer to the Moot, albeit one completely tilted in his favor, at the conclusion of the Stormcloak quest line.
Pelagius III was awarded the throne of Haafingar by Emperor Cephorus because of his noble blood line, his participation in the Siege for Solitude during the War of the Red Diamond, and because there was an <ahem> sudden vacancy in the throne at the conclusion of the Siege. With the conclusion of the Civil War, that ship has sailed for the Dovahkiin so there's no convenient vehicle to his ascension to Jarldom let alone the throne of the High King. Metaphorically speaking he's the teenager stuck at the kids' Thanksgiving table (Thanes) until someone at the adult table dies.
.... Also, that same Wiki you quoted has him listed as Imperial, Nord, AND Altmer, but not Breton...
Okay, you just equated UESP to The Elder Scrolls Wiki which almost made me throw up in my mouth a little. There is a huge disparity between the accuracy and reliability of the lore articles of those two sites as the quality of the articles at the latter are mediocre and filled with a considerable amount of inaccuracies and pure fabrications based on biased speculations and ignorance that have no place in a legitimate wiki site.
Pelagius III was indisputably Breton. This is irrefutably established by the lore. The Breton bloodline was introduced into his bloodline by his grandmother Quintella of Camlorn who married Pelagius II. Her son and Pelagius III's father was Magnus, a Breton by race (it's established in the lore that the progeny of couplings of different playable races in the Elder Scrolls don't produce hybrids but rather offspring of one race or the other of the parents). This is further confirmed by the fact that Cassynder, his son with his Dunmer wife, Empress Kytariah is also Breton. Finally as in-game proof, you meet him during one of the Daedric quests in Skyrim and the console shows his race as Breton.
Regardless, the player character is a very poor choice for High King. It's inconsistent to the lore to place the main character of an ES game in such a prominent political position. It would be awkward at best to have future lore documents refer to the High King of Skyrim after the Civil War to be a nameless genderless person when every other person of prominence in a lore document is referred to by name. It's also inconsistent to do so in situation which would have to be based by explicitly choosing one of two conflicted outcomes based on player decision over the other (i.e. the Civil War faction quests) as an explicit part of the lore. That would be unprecedented and, aside from messing up the lore of the game series, it would bear the potential of alienating a substantial part of the fan/consumer base of the game series.
As a aside, pointing to all the various guilds the Dovahkiin leads and the Thane titles he or she holds is completely unpersuasive and inapplicable for advocating such a thing because the lore never attributes the side quests to the main character in ES games. In the lore those accomplishments are attributed in a vague manner to another person who is never identified with the main character.
