It's possible, maybe likely, the PD during TOS isn't that old. In "A Private Little War" Kirk mentions recommending a "hands off" policy regarding Neural thirteen years prior. It's sketchy, but just maybe in the early days invoking the PD wasn't necessarily automatic and depended on individual circumstance. And as has been said before it seems more nuanced and layered, less absolute, than during the TNG era.
At certain times evidently the PD can be waived---a notable example would be regarding Organia. And/or the PD might not apply when a culture was contacted before the PD was adopted such as with the Cereans in "Friday's Child." Note Akaar's references to Earth men suggesting they've been familar with them for quite some time.
So for all we know the PD might be only about twenty years old (give or take) during TOS.
This might also dovetail with the age of the Federation. There are references (such as in "Whom Gods Destroy") and subtext that suggest the Federation might only be a couple of decades old at this point. It's never spelled out in TOS because they never thought it important, and the third and final season was winding down, but the subtext is there.
And wasn't there a reference to conflict with the Klingons going back about fifty years? I know there's one in TUC mentioning seventy years of conflict with the Klingons. This certainly flies in the face of Picard's reference in TNG of contact going back two centuries.
Back to more on-point. Kirk initially didn't set out to destabilize the society of Beta III. They went down in native costume and tried to gather information incognito. When they were discovered Kirk tries twice to reason with "Landru." If the Landru computer had been more flexible it could have agreed to release the Enterprise
and allow Kirk and company to be on their way. But it was rigid and could only see the option of destroying whatever it decided to be any contaminating influence. Kirk's use of logic was the only recourse to get the computer to shut itself down. Of course, by the rules of '60's conventions that meant the computer frying itself (with pyrotechnics) rather than simply getting stuck in "blue screen" or "spinning beach ball of death" mode as would be the case today. Then it could be reprogrammed, if desired.
The PD supposedly doesn't apply to stagnant societies. If indeed the culture of Beta III was stagnant for thousands of years then Kirk didn't violate the PD while saving themselves and the ship. And Sawyer is wrong.
Another interesting point is how advanced the original society must have been. It's best evidenced in the fact "Landru" could neutralize the landing party's phasers and mostly that it had access to means to destroy something in orbit. That suggests the original culture had reason to believe danger could come from beyond their own planet. The same applies to the Val computer in "The Apple."