...And as I've also noted in an earlier thread: It's easy to read this as Kirk not falling in love with anybody at all, but merely executing a clever campaign to defeat his foe. Just like he always does with the men and women he meets.
Spock and McCoy really accomplish nothing here. Their lines of inquiry into the mystery of why Flint isn't helping them, as well as their attempts to overcome Flint's obstructionism, are seemingly logical but miss the mark completely. Kirk in turn cuts right to the chase, understanding what makes Flint tick and hitting him in the soft (Flint, unlike M-4, is quite defeatable in fisticuffs, and Rayna can be swept off her feet with basic techniques Kirk is well versed in) and sensitives (Rayna is a major weakness in Flint's armor).
In the end, Kirk weeps. But not for Rayna, at least not by his own words. He's weeping for Flint, for whom he apparently didn't feel erotic love, but whose fate is a sorry one, and in many ways a painfully close match of Kirk's own lot in life.