I think the correct way to represent this is as a time series - the past states of the plant cause future states of the plant, and also have a causal effect on future states of the soil.

That presumes discrete time. But time is continuous. (Speculations about discreteness on the scale of Planck time are irrelevant to the timescale of plant growth.) Any discretisation involves an arbitrary choice of time step. How do you make that choice? What can you do with a causal diagram constructed in this way, with millions or billions of nodes? With an assumption about the invariance of causal influences over time, it can be represented in a compressed form in which only two time points appear, but it's not clear to me that that offers any advantage over cyclic diagrams and continuous time.

The structure is kept somewhat simple by the fact that time steps only cause the very next time step - when predicting the future, knowing the present state of the world is enough (if you're Laplace's demon).

Only if the "present state" is defined to include all derivatives of the variables you're interested in (or as many as are causally relevant). Computing (a discrete approximation to) the nth derivative of a variable in discretised time requires knowing the value of the variable at n+1 consecutive time points.