An example (Hartshorne Ch II Ex 3.7) where the condition at the generic point is not vacuous is:
Suppose that f:X -> Y is a dominant finite type morphism of integral schemes such that Y is irreducible and the fibre over the generic point of Y is finite. Then there exists an open dense subscheme U of Y such that f: f^{-1}(U) -> U is finite.
One can also check flatness over a curve by checking whether certain points get mapped to the generic point.
"Generic vanishing" also holds for coherent sheaves in the sense that a torsion sheaf is defined as one which is not supported at the generic point.
I guess one point of view on the philosophy is that with appropriate finiteness conditions good properties at a point should extend to a neighbourhood. In particular, good properties at the generic point should extend to a dense neighbourhood.