Why not develop a Hamiltonian-based Morse theory?

Question

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory am curious why the seemingly more natural analogue of this can not be applied to the study of symplectic manifolds (or indeed maybe it can but is of no use?). By this I mean a sort of finite-dimensional "Hamiltonian Morse theory" where the gradient-like vector field is replaced with a Hamiltonian-gradient-like vector field $X_H: i_{X_H} \omega = dH$, and it is the critical points of $H$ that are relevant rather than those of the action functional (as in Floer homology). Any comments would be welcomed.