I think the only time Kirk actually broke the PD was in "Miri", when McCoy's antitidote to the virus was not only given to the Enterprise crewmembers, but to the children. Strictly speaking, the children shouldn't have been treated. It would've made for an interesting debate within the episode, but I don't think any writers had created the PD, yet.
By the 24th-century interpretation of the PD, where intervention is forbidden even if it means letting the civilization die off, you might be right. I don't think that's true of the 23rd-century interpretation, though. As Kirk pointed out on various occasions, the PD was about allowing the normal, healthy development of other cultures. That meant not interfering or imposing the Federation's will on naturally developing cultures, but Kirk also interpreted it to mean freeing stagnated or oppressed cultures from the influences that kept them from healthy development. No one could say Miri's culture was healthy, or even that it was a viable culture. The disease would've killed them all off eventually, if famine and other illnesses didn't do them in first.
I'd say Kirk's one blatant violation of the Prime Directive was preventing the execution of Eleen and her unborn child in "Friday's Child." That execution was rightful by the customs of her people, and she herself accepted it. Now, if it had been Kras the Klingon who'd killed Akaar, or enabled Maab to do so, maybe it could be argued that Kirk was countering Klingon intervention. But to all indications, Maab himself instigated the coup -- perhaps egged on by Kras, but it was pretty clear that the reason Kras backed him was because Maab already had designs to seize power. And it was Maab's forces who killed Akaar while Kras was confronting Kirk. So the whole thing was kosher by Capellan law and custom, and Kirk had no basis for intervention on Prime Directive grounds. He was forcibly imposing his own morals onto members of an alien culture in defiance of their established customs. That's about as blatant as a Prime Directive violation can get.
A lot of people cite "A Taste of Armageddon" as a PD violation, but I don't, since the Eminians had by that point attacked the Enterprise
and declared their intention to kill its crew -- essentially a declaration of war against the Federation. The PD doesn't forbid taking action against a declared enemy in wartime.
EDIT: TNG: "Angel One" alleged that the Prime Directive doesn't apply to civilians, but I've always found that an odd interpretation. Nikolai Rozhenko was definitely a civilian; he entered Starfleet Academy, but soon dropped out.