I'd rather argue he was a specialist who knew where his strengths lay. Picard seemed to know a little bit about everything - a true mark of the Google generation, and no longer quite as noteworthy as in the 15th century Europe... In contrast, Kirk had specific hobbies, such as 19th century United States history, but he didn't appear all that diverse.
His roots seem to lie firmly in the skipper of Forbidden Planet. He's more likely to take pride in his ignorance and caveman ways than be ashamed of it. Doesn't mean he wouldn't have intelligence or education or knowledge, or a combination thereof. He just doesn't flaunt it much in TOS.
We have to remember, though, that the retired Kirk was able to readjust the deflector machinery of a brand-new starship in ST:GEN. It's really unlikely that an average individual would have possessed that specific skill... Either Kirk happened to have made a really fortunate choice of retirement hobbies, or then he really had more under his wig than met the eye.
Timo Saloniemi