Great point about A Private Little War.
Kirk does sort of tell them that the Klingons are a threat, though, in a very Kirkian, polite-even-to-enemies manner. And the "small scout vessel" was evidently no match for (or wanted no part of) the Enterprise.
Both "Friday's Child" and "A Private Little War" also showcase something different about the 23rd century United Federation of Planets. In both of those episodes it was implied that the Prime Directive wasn't automatically applied.
Dr McCoy did a cultural planet survey in "Friday's Child", and Kirk did one in "A Private Little War", and it was after the latter survey was submitted that the Federation (where Kirk himself concluded in the report that the planet should be subject to the PD), invoked the Prime Directive for the planet Neural.
In McCoy's case obviously the Federation chose not to invoke the Prime Directive for Capella IV, probably because of the fact the mineral that was abundant on the planet was of use to the Federation.
And even in "The Omega Glory", it's interesting that Kirk and Co. technically find Captain Tracy in violation of the Prime Directive, when Captain Tracy himself states that as this was a new world he stayed behind after the initial landing party returned to the USS Exeter "to arrange for the planet survey with the village elders"...
I think it's more a case that technically, when making first contact with a new world, The Prime Directive temporarily always applies until a survey can be conducted to see if enforcement of it for the new planet in question should be maintained.
The above said it's kind of interesting in that to do the planet survey, The Federation representative in question has to break the Prime Directive to arrange for the planet survey. Remember that Tyree knew all about Kirk, and the Federation; but promised silence after Kirk left.
^^^
If that's the Federation's SOP in the 23rd century, what happens if the person or persons who are aware of the Federation representative's true origin decide (for whatever reason), to break their promise and reveal that their Planet was visited and surveyed by a being or beings from another world?
So yeah, In those terms the whole execution of the Prime Directive in the 23rd century has some interesting permutations.