Kirk’s Empathy
Kirk will and does go to great lengths for those he cares about, and that empathy is what makes Kirk an interesting character.
He is at his best when he’s fighting against long odds to help those he feels responsible for, because it’s when he stops being an action hero and starts being a guy you root for. His empathy is what connects you into the story, because you want him to succeed in his efforts to protect the other characters.
The flip side is Kirk’s empathy for those he’s just met. There are countless episodes where the crew discovers some enslaved group or oppressed minority or hoodwinked populace and Kirk just wants them to be free. His desire for everyone everywhere to have control of their own destiny is the motive force that drives the series, and it reflects the core ideals behind the show: that sixties-America binge of freedom as unadulterated good, as the axis around which everything turns.
That Kirk is a starship captain is the most fitting piece of the puzzle; he is freedom incarnate, zooming around the galaxy doing things that he wants to do because he thinks that doing them is awesome. That his adventures so often find him freeing people from bondage or escaping bondage himself is part and parcel of the enterprise.