That is simply a corollary of the fact that Pearson's r and regressions are usually linear/additive, and things like Meehl's demonstration wouldn't work if they weren't. You'd just calculate all the pairwise correlations and get nothing if they were solely totally nonlinear/interactions. (In which case you'd have a hard time proving they were related at all.)

> You'd just calculate all the pairwise correlations and get nothing if they were solely totally nonlinear/interactions.

I don't believe this. Most nonlinear correlations also show up as non-zero (linear) correlation coefficients. There are really only a couple pathological cases I can think of where it would not happen.