Let me try to explain another way. Kirk doesn't come across as an everyday guy to me. He presents as someone who has the answers, and is largely unchanged or unaffected by the end of the story.
He reminds me more of Superman than he does Luke Skywalker.
I think we mean different things by "everyman." My point isn't that he's some schlub. My point is that the Trek universe is fundamentally one where hard work and determination matter more than fate. Anyone can be a model Starfleet officer if they put their mind to it. That was the whole friggin point of Nog's arc in DS9 - he climbed up from a juvenile delinquent to a dedicated Starfleet officer.
Kirk isn't a superhero. But he certainly IS a larger-than-life hero.
He has no superpowers, but he is way above the curve in every aspect. Not as much as Batman, nor Sherlock Holmes (who clearly dip into "superhero" territory). But he probably comes close to James Bond, Flash Gordon and a lot of other pulp- and noir-heroes. Maybe like Robin Hood, who's an enourmously good archer - not superhuman good, but very much extraordinary, and that in many different regards.
I think what stands out about Kirk, is that none of his abilities are unrealistic - every single one can be achieved through practice and training. It's only the sheer number in which he is an expert that is uncanny.