I was reading Schwartz's QFT book, and in Chapter 30, he introduces the calculations of anomalies by evaluating objects like $\partial_\mu\langle J^{\mu 5}J^\nu J^\alpha\rangle$, where $J^5$ is understood to be the axial current, whereas $J$ is the usual vector current. It's clear to me that this correlation function will yield triangle diagrams with amputated external legs and it's also clear that, should the theory be anomaly free (in this particular case free of chiral anomalies), this object should vanish.
What is not clear to me is why are we looking at it in the first place. Why should we look at this and not directly at, for example, $\partial_\mu\langle J^{\mu 5} \rangle$, if the anomaly is in fact there, shouldn't this, at some loop order, yield a non-zero value and that's that? Isn't this precisely what is done when we calculate it via the Fujikawa method? Why is it different when we look diagrammatically?
Also as an aside, which is related, but not the same topic, the book states that seeing non-zero amplitude related to the $\pi^0\to\gamma\gamma$ decay is equivalent to stating that the composite operator coupled to the pion has a non-zero expectation value in the presence of a background EM field. The book says he'll derive it rigorously later on, but I don't even understand the intuition behind it, if someone could give me a brief holistic argument as to why this is the case I'd appreciate it as well.
