"They must apply to everyone, or they mean nothing!" strikes me as a misplaced rhetorical flourish. It sounds exciting and forceful, but it doesn't fit the situation.
Second, taking Kirk literally implies that the Yangs now have a moral duty to conquer all Comb territory and annex it under the government defined in the Yang Constitution. And that doesn't sit right as a moral imperative.
You´re right it is a badly written line, for exactly that reason. What Kirk meant to say however, IMHO, is that the rights
granted to the Yangs must also be granted to the Comms. That the good, the justice and everything else proclaimed in the cited text should be available to everyone, not just to your own people but also to those wo think differently from you.
Like so often in Star Trek this can again be applied to the real world. And written during the time of the Cold War gave it an extra and very current meaning (and political explosiveness?).