Why is any lumped parameter system modelled with an ODE?
Lumped parameter systems are systems that contain first-order integrators with respect to time (capacitors), first-order differentiators with respect to time (inductors), and multipliers (resistors).

It is immediately trivial to convert a first-order ordinary integral equation into a first-order ordinary differential equation, by differentiating both sides with respect to time.

At that point, the first-order ODE is the natural representation.

The state vector x contains the state variables for each integrator and differentiator.  For a capacitor, the state variable is the voltage across the capacitor.  For an inductor, the state variable is the current through the inductor.

Note that you can "stack" first-order "stages" to get Nth-order systems.

It is really a shame that analog computers have gone out of vogue.  Seeing the elements stack up is a really educational experience.