Let me play devil's advocate here: I'm not sure that I agree that the ubiquity of π is so mysterious. After all, how do you ever prove that π appears? You have to relate your situation to some known situation where π already appears, so the mystery is solved almost before it occurs. Of course, if you're just shown the appearance of π without the proof then you may be surprised, but that simply means you haven't yet seen the proof. To take an example, the proof that $\int_{-\infty}^\infty e^{-x^2}dx$ involves π uses the rotational invariance of the normal distribution. But rotations are closely connected with circles and hence with π, so it isn't too surprising that π shows up. To take another example, it is amazing that $\sum n^{-2}=\pi^2/6$, but one nice proof of that uses Parseval's identity, calculating the $\ell_2$ and $L_2$ norms of the Fourier coefficients of a certain function and the function itself, respectively. And Fourier coefficients involve trigonometric functions, so the appearance of π is, once again, not a surprise.
Maybe the right thing to say is that the multiple appearances of π are initially striking and mysterious, but the mystery disappears on closer inspection -- like many mysteries. The statement I would dispute is that there is a general mystery of the kind "Why does π appear so much?" I'd give an answer like "Because circles and rotations appear a lot."
