Implementations of HMC are approximating ideal Hamiltonian dynamics by discretizing time. The "symplectic" integrators that are used have the property that they

Implementations of HMC are approximating ideal Hamiltonian dynamics by discretizing time. The "symplectic" integrators that are used have the property that they correspond to simulating continuous-time Hamiltonian dynamics exactly on some other energy surface, which is not too far from the one you had in mind. So, in some sense, an approximation like the grandparent had in mind is the usual case. You are right that a bad approximation of the surface could lead to proposals that get rejected a lot. But at least the dynamics are likely to be stable. It could be worth trying.