With the Klingons involved as well in "Friday's Child", I think the Federation would be looking at a larger political landscape.
Frankly, it came down to getting a mining agreement with these people and having them be independent but friendly to the Federation, or having them fall into Klingon hands or at least be a Klingon minion. In Klingon hands no laws in that culture would be respected. Indeed, the entire culture may be stifled. What Kirk did to try to avoid the Klingons getting their way on the planet was minor compared to that. It may not have been his shining moment, but occasional moral and legal ambivalence was a hallmark of the Cold War.
I don't think that when Kirk saved Eleen, he was thinking "I need to ensure this baby is born so that there is a legitimate claimant to the throne and Maab's claim will be invalidated." I think he was thinking "I can't let them execute a pregnant woman because that's just wrong." He was imposing his personal and cultural morals onto a society with different morals -- and specifically he was forcing rescue on Eleen herself, who wanted no part of it. Of course personally I agree with his position, but the Prime Directive means respecting other cultures' right to their own laws and customs even when you find them abhorrent.
This is what I mean by the TOS version of the PD versus the TNG version as while the TNG version made it apparent that what Kirk did would be a violation of the Prime Directive, the TOS version only seems to cover not screwing with pre-spaceflight planets, they never said anything about it affecting dealings with species they already made contact with who the majority of which knows they're from outer space.