I still don't understand why people find it so unrealistic that Kirk was promoted from Lt to Captain? No one even commented on me pointing out the President of the United states. He was a Community Service Guy, and got elected to State Senate, and spent almost the entire time running for Federal Senate (Bypassed most votes by voting "present") and then he left the State Senate seat before his term expired, and did the same thing with his Federal Senate Seat, spent most of that term running for President, skipping most votes by voting "Present" and left that seat before his term ended in order to be President. So, he didn't even finish his Academy training, and he was given the most powerful position in the world, with no experience/training other than running for office. Then, once in office, within a month of taking office, before he had time to do anything except unpack, he was given the Nobel Peace Prize, and has gone on to continue the Policies he railed against as a Candidate. With a real life example like that, how can anyone possibly argue that Kirk's promotion is unbelievable and unrealistic, and it is a fictional story, not real life?
Because Politicians (in any world) aren't analagous to Starship captains? A politician in essence is nothing more and nothing less than some guy who, through whatever reasons, proves capable of convincing a whole lot of people that he should be in charge. There is no 'Academy' for politicians, and no matter how many of them you can point to who all got into the game through more or less similar paths, there are always other examples who did not. Ultimately, their only legitimacy lies in the election results, and electoral candidacy is open to everyone with only a very minimal number of actual requirements.
Starship captains, on the other hand, are appointed by a single regulatory body which is presumedly bound to specific rules and regulations regarding command suitability, etc, etc. Except in societies where corruption is widespread enough to render these rules moot in the face of favoritism or bribery - which is clearly not the Federation - Captaincies are not just handed out to anyone who does something impressive without some sort of actual promotion process.
You apparently aren't very familiar with US Politics. While it has the face of Democracy, and it's true the people vote from the Nominees, it's not true that all you need to do is decide to run for President. In order to have a chance at winning the Nomination, you need to impress someone within the Party Machine and have them advocate for you, the party machine pretty much decides who they want to run in the election against the other party, many have been told "It's not your turn, drop out" or simply didn't have a shot because the Party Machine didn't want them, they had reasons for wanting someone else. Candidate Obama impressing someone in the Democratic Party Machine who advocated for him, is no different than Captain Pike Advocating for Kirk (Remember Hillary Clinton was pretty much a sure thing to win the Nomination, but, the party Machine decided they wanted the First Black President, instead of the first Female President, and swung the nomination votes to Barack Obama instead). Also, Admiral Marcus told us in the movie that he liked the cut of Kirk's Giblets and wanted him in that seat, because he thought Kirk was the kind of loose cannon who would go along easily his plan to start a war with the Klingons. So, Kirk won against the Kobayashi Maru (an unwinnable scenario) with creative thinking, saved the Earth and impressed a well respected Captain and an Admiral who advocated for him to be put in that position.
As far as President Obama's prior Political career being analogous to the Academy, in order to be qualified to run for President, people expect you to have some Foreign Policy Experience, or a voting history they agree with or Management experience (CEO of Corp, General in the Military, Department Head of Large Gov't Segment, High ranking member of important Government Committee(s), Party Leader, State Governor, etc). So, yes, there absolutely is an analogy for President Obama not having completed his "Academy Training".