That presumes discrete time. But time is continuous.

Yup - any discrete causal model is an approximation. As with any approximation, one chooses it based on what you can exactly solve, what you have the resources to calculate, and what kind of things you need to calculate.

Only if the "present state" is defined to include all derivatives of the variables you're interested in

Indeed - the classical world actually lives in phase space. Quantum mechanics is actually somewhat simpler that way.