I understand the Laplace transform ideas above (used to TA the class where it's taught) and I agree with your that they are more definitional than explanatory.

To provide actual explanatory power, you'd have to explain what the reactive portion ("imaginary part" in the phasor notation) of the impedance is actually doing in real terms.

What it's doing is to cause the current to lead or lag the voltage in the AC signal. In a purely resistive circuit, current and voltage vary exactly the same way at every instant.

You can actually go a long way with this just by understanding that, e.g., for an inductor,

V = L dI/dt

so that small current changes will imply very large voltage changes. If you're in to intuition, this means "voltage leads current" (i.e., the sinusoid of voltage is 90 degrees ahead of the sinusoid of current).

The above equation is also why lights in your house dim when you turn on a motor (= inductor). There is a significant current change, and the voltage changes even more, so the lights dim.

Or, you can be even more intuitive, and note that it's obvious that when you put a coil of wire (= motor = inductor) across the two sides of your house's electrical wiring, the coil of wire will pull the + and - conductors together in voltage at first. Until the magnetic field sets up to provide reactance, the coil of wire is basically a short circuit.

And finally, a capacitor is just the dual of the inductor.