I don't recall any health concerns for Kirk in the Primeverse.
There was definitely an episode where McCoy was close to suggesting a diet for Kirk. Lots of tie-in novel authors picked up on that stray comment over the years and, when Richard Arnold was vetting the tie-in manuscripts in the late 80s/early 90s, he specifically stopped them from doing so.
McCoy expressed concern of Kirk's mental health in "Obsession".
In at least one version of TMP, McCoy mentions "command fitness".
if he does have some sort of terminal or even potentially fatal illness
Huh? You're reading way too much into the scene.