Tracey had already "contaminated" the planet by taking sides with the Comms and using phasers as well as arming the Comms with the phasers he took from Kirk's party. Advanced tech had already been introduced. Kirk "explaining" the meaning of the Yangs' document hardly complicates matters. If anything it might prevent further bloodshed...but I doubt it. And how do we know the natives hadn't already seen Tracey's original landing party materialize out of thin air and then disappear later as they went back to the Exeter.
I can see lots of dissent over what the document is supposed to mean after Kirk and party depart. Note that when the original American Constitution was drafted slavery was in effect in the U.S. Certainly looks like the founding fathers weren't including those enslaved as being equal with everyone else. Extrapolating from that I imagine there would be lots of Yangs insisting their holiest of holies can't possibly apply to the Comms as it does to them.
I think that technically, and truthfully as well as in spirit....Spock is quite correct whenever he tells Kirk, that he is violating the PD. It's Spock after all! Come on!
But I can also easily see the "Prime Directive Board" (which is probably human-dominated) rolling their eyes at the Vulcan contingent and telling Kirk, "Wellllll...no harm was done. Off with Ye then youuuuuu. Get out of here you knucklehead! Why I oughta...."
I also noticed that in the beginning of the episode, Kirk was going to let Cloud William be executed. ...of course he couldn't restrain himself when the more fetching Julie Newmar was going to be executed in "Friday's Child".
edit: Oh, and in direct response to your second point...so Kirk caused a Civil War among the Yangs? WTG Kirk!