The triumvirate's relationship is tested when the unique alien biology associated with Spock's ancestry sends him spiraling into a destructive, quasi-pubescent madness lest he is reunited with his home planet.
"Returning to The Source" is a classic mythological motif representative of thematic conflicts beyond the raw science fiction conceit on display here. While biologically interesting, and precedented with a cunning display of techno-babble (salmon's natal homing), the true core of this concept is its thematic focus on the nature of Spock's (and by extent, all Vulcans') internal conflict:
The extremity of his dedication to logic and reason must come with a cost. Reproductive necessity escapses no one, though we may wish to think it does when considering those steely-exteriored, Spock-like figures in our own lives. Vulcan is an entire society extrapolated of the unbalanced attitudes present in our own.
The barbaric nature of the Vulcan rituals depicted calls to mind images of familiarly anachronistic traditions kept alive in our own society. Traditions also birthed by an imbalanced reconciliation of our base humanity. The extremes of Vulcan's ritual answers to base questions of existence mirror our own historical barbarism and presently clinging traditions.
This is dramatically presented in contrast to the comparatively balanced (read: superior) approach of the Federation. Specific note is made of the Vulcan leader's lone reputation for denying a seat on the council. On the macro, this universe-building definition of conflict appropriately reflects the micro drama of the character conflict. It's what this episode is known for: Kirk vs. Spock.
But what, exactly, does the movement of the story say in the way of theme? What is it that breaks Spock from his impassioned blood rage? The exposing of this ritual madness to that very presence of an outsider. Through the contrast, a mutual understanding is born. This is directly in line with the very core of the Kirk/Spock/McCoy dynamic: the reconciliation of opposites. Just as Kirk is Action tempered by Logic (Spock) and Passion (McCoy.)
The inherent balance of Spock's own "outsider" status on Vulcan -- as a legendary officer of the Federation -- and a particularly illogical (and therefore distinctly non-Vulcan) conclusion brings Spock to a level of understanding that allows him to transcend the conflict both internally and externally:
You may find that having is not so pleasing a thing as wanting. This is not logical, but it is often true.
One can't help but feel the entire motivation for the situation's creation was to put our leads at odds. However, the exceptional dramatizing of a multi-level conflict and resolution alone justifies its existence. Its effectiveness is undeniable considering the ensuing reputation and cultural impact. (Who can forget that music?!)
On the surface, we're given a story about the experience of seeking out "new life" and the exoticism of foreign culture. The stakes are raised by supporting universal conflict and represented theme. For that, "Amok Time" has earned its place next to the greatest episodes in Star Trek's history.