I've only ever causally watched TOS, but having recently rewatched the Star Trek II, III & IV movies, which you recall encompass an extended story-arc, I'd say that the Kirk-Spock relationship was absolutely core: one without the other would not simply half the impact, it would decimate it.
Their friendship, reinforced by Spock's self-sacrifice and Kirk's loss (of his son), culminates in the the success of Kirk's Orphean Katabasis. It is only the raw sting of grief that sparks Kirk's anagnorisis, which in turns allows him to cast aside his vanity (his cool demeanour) and status (his rank and command) as mere trappings of life.
Kirk's humanity may be flawed (which is mirrored by the human scientists' use of unstable proto-matter), yet the presumption of Man in attempting to emulate divine Genesis and overcome death is ultimately rewarded owing to the beauty of his 'song' (Kirk's-Spock's-our shared story).
Kirk: [looking up from the planet surface to see the remains of the Enterprise burning up in the atmosphere] My God Bones, what have I done?
McCoy: What you had to do, what you always do. Turned death into a fighting chance to live.
Even once Hades and Persephone have released Spock to return to the living, it is his memory of his friendship with Jim that proves the key to regaining his identity.
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And what is most sci-fi today? CG explosions and bloody action figures.