Spock did reach the rank of Captain while teaching at Starfleet Academy. He then goes onto become an Ambassador who leads the reunification efforts between Vulcan and Romulus. So Spock was perfectly fine being a leader but I think it's also fair to say he didn't want to Captain a Starship. He did his leading in his later years too when he was fully at peace with his Human half. So yeah, I'm going to say you're spot on in saying it had to do with the social aspects.That's an interesting interpretation. I also picked up on that watching the reruns as a child, but my assumption was that Spock was an introvert, like me, and didn't have interest in the social aspect of being a captain. He was happy with his indirect influence over leadership as a trade off for not wanting to be in the spotlight all the time.
There's also a gap between the intended portrayal of James Kirk and the actual portrayal. One that got exacerbated in the Kelvin timeline movies. Kirk was a great tactician, with strong instincts, and his ship and his mission were always top of mind. Most of his fighting and seducing was just a means to completing the mission. And because 60's TV we don't get to see the missions where he makes diplomatic solutions because the execs wanted babes and action.

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