Let $X$ be a smooth projective variety over $k$. Variety here meaning a separated geometrically integral $k$-scheme of finite type. Is the structure morphism $f : X \to \text{Spec}\;k$ flat?
I guess this means we should check that for any $x \in X$ the induced map $f_x : k \to \mathcal O_{X,x}$ is flat, so for any $x \in X$ the stalk of the structure sheaf should be a flat $k$-module. But then a $k$-module is just a $k$-vector space, which is always flat. Am I missing something?
Your proof is correct (as was already confirmed by others in the comments).
A little picture: heuristically, a morphism is flat if all the fibers look the same, and for projective varieties this heuristic is a theorem (flat iff all the hilbert polynomials of all the fibers are the same).
Over a field, there's only one fiber, so they tautologically "all look the same."