Well, I too, would love to see more of ape-Buddha, AKA Dr. Zaius's great great grandfather. He's a cool guy, certainly not fully explored. I'd disagree that Caesar's worrying about being alpha, he's more concerned that his next-in-line is gonna be destructive not only for humans but also for the apes (hmm, now where have I seen that recently?).
I think you're right that this moment didn't necessarily have to be the turning point, though I think the movie justifies its importance in two ways. Firstly, working backwards, the conclusion sets up the ape vs. military battle that is to come, a clear escalation from this settlement (albeit defended with military tools) the apes tussled with in the current film. This is the moment that leads to that, so it's important. Secondly, and I think more interestingly and importantly, that escalation comes out of misunderstandings and prejudices, the stuff of all wars in one way or another. The film clearly outlines how both sides gear up for war through rhetoric and racial othering (again, sound familiar?). It's a fantastic depiction of that process, with sympathetic characters all around, including the villains to some degree. The human fear is understandable, they've lost everything so far and are just trying to hold on to what they've got. The apes fear the same prosecution and violence they've all experienced from humans before things went to shit. It's understandable on both sides.
I think this movie is indeed about human history and ape history. It deals with the aftermath of animal testing and, shall we say, incarceration? It also deals with the history of humanity in the constant warring between factions based mostly on the idea that they are different in one way or another. This is that history, it's just rendered into a human/ape-scale. That's part of why people called it Shakespearean. It's what Shakespeare did, he wrestled with big ideas through the human characters that represented ideas and drama greater than themselves and their interactions. It just happens to also have an ape riding in a tank.