The key to the original question is the askers mention that it was always met in wave behavior situations-periodic functional solutions. The periodic behavior is a strong constraint place upon two paired variable which describe the system behavior. Some examples are voltage-current, $E-B$ component of electromagnetic fields, elastomechanical motion force-velocity. Generaly the product of the pair yields power (energy/time), but are constrained such that the ratio of the pair is at-most a complex constant-(the impedance or its reciprocal). Periodic functions have this property.
One responder suggest Force-velocity as a pair, but without qualification. Simple counter example is provided by a constant force. $Z = \frac{F}{v} \sim \frac{1}{t^{2}}$. One might argue that it is true at least at $t = 0$, but at initial condition $f$,$v$ are uncoupled and independent. The utility of impedance lies in its constancy of proportionality.
The arguments the utility of impedance in power transfer problems misconstrued several major points by all sides. One often overlooked property of $Z$ is that it is periodicity dependent (i.e. on the wave frequency). Maximum power transfer occurs for source-load impedance match only over a limited bandwidth because generally the source and the load impedances vary differently with frequency. Match device must be retuned for large frequency change or else the matching range has to be increased with complex structures. Fine speakers and radio transmission line often get quite complex.
The arguments above also conflated harmonic balancing with maximum-transfer problem. They are intertwined but not identical. In a high fidelity systems, signal distortion almost always trumps the power efficiency issues. Thus eliminating distortion by reshaping the energy balance over the required transmission waveband is regularly practiced by shaping the $Z \left( f \right)$ distribution - i.e. by filtering.
Impedance is a useful concept when it expresses a constant (in time but not necessarily with wave-frequency) relationship between a pair of variables that characterize the behavior of a physical system. The product of these variables is the power transferred by the wave (or of their integral or derivative). Periodic functions satisfy the constant ratio constraint - Most non-periodic functions do not.