Small-signal analysis of inductor/capacitor
Your analysis is wrong.

Small signal models consist in a linearization of the non-linear equations describing a certain device (as a transistor or a diode). Of course, it makes sense only if such device is non-linear, i.e. described by a set of non-linear equations.

It is called small-signal analysis (and the models are called small-signal models) because they are only accurate if the signals you're considering have "small" amplitude (as the linear approximation of a non-linear equation can be used to make accurate calculations only if you consider quantities nearby the point around which your analysis is centered). Read about Taylor series for further info.

The reason for the linearization of a non-linear equation: the second is a lot harder to handle and/or to solve.

The equation you wrote for the capacitor and the inductor are already linear, i.e. there's no need for linearization.