> Either way, I can get arbitrarily good approximations of arbitrary nonlinear differential/difference equations using only linear probabilistic evolution at the cost of a (much) larger state space.

This is impossible. When driven by a sinusoid, a linear system will only ever output a sinusoid with exactly the same frequency but a different amplitude and phase regardless of how many states you give it. A non-linear system can change the frequency or output multiple frequencies.

As far as I understand, the terminology says "linear" but means compositions of affine (with cutoffs etc). That gives you arbitrary polynomials and piecewise affine, which are dense in most classes of interest.

Of course, in practice you don't actually get arbitrary degree polynomials but some finite degree, so the approximation might still be quite bad or inefficient.