From a numerical standpoint, although you can interpret SDEs as non-autonomous ODEs, the forcing term is not differentiable (it's "almost" Holder continuous $\alpha=1/2$ a.s.) and thus the error estimates one uses to derive numerical methods for ODEs usually do not apply (for example, Runge-Kutta order $k$ methods assume $k$ derivatives in their derivations). In fact, in just about any case where $g$ is not a constant, the order of convergence for deterministic methods on SODEs ends up being $1/2$, which is clearly sub-optimal. Thus instead the numerical methods for SODEs have to use the properties of the "Ito derivative" and stochastic Taylor series to derive new methods for which the higher orders of the error analyses apply.

However, what you pointed out is where a lot of the intuition comes from. If you read work from researchers like Hairer or Kloden, they frequently build the heuristics for understanding the system by approaching them as non-autonomous dynamical systems where the forcing term has not so nice qualities (i.e. Holder $1/2$ or in the case of space-time white noise, $1/4$). Thus in some sense you can understand things heuristically like "the deterministic methods are only order $1/2$ because the forcing term is Holder continuous $1/2$", and so understanding the SDE as a type of ODE has value in that sense.