I can certainly see where you're coming from, I have the same problem with many of my physics courses. Too much mathematical formalism and not nearly enough talking about actual physical systems and intuition.

> I guess my point is, formalisms seem obvious once you understand the concept intuitively, but are a terrible way to gain that intuition. The basic idea of resistance is simple (a resistor restricts the flow of electricity), reactance was new to me but I got a rough idea that it similarly resists the flow of electricity in an AC circuit and is a function of the frequency.
> So if I know that the impedance is a right-triangle combination of those two things, I've got a great start for getting some intuition: I know that anything that has resistance or reactance also has impedance, and something that has both has even more impedance. The formal definition tells me none of this.

This is a rather superficial understanding of impedance. Since impedance is a complex quantity, "more impedance" doesn't really make sense unless you define an order on complex numbers. Reactance is much more complicated than resistance, as you say because it is a function of frequency instead of one number. Impedance has the same difficulty: it is a function of complex frequency s.

Suppose we have an electrical element with impedance Z(s), then we have V(s) = Z(s)I(s). The problem with this equation is that it is an equation in the complex frequency domain, just like you get equations in the ordinary frequency domain with Fourier transforms. What we do with Fourier transforms is to look at what happens at a particular frequency omega. For example if you had V(omega) = Z(omega)I(omega), then you'd know that if you applied a sine wave with frequency omega current, you'd get a sine wave scaled by Z(omega) voltage. Or equivalently, if you applied I(t) = e^(i omega t) current, you'd get V(t) = Z(omega) e^(i omega t) voltage. Laplace transforms just generalize this complex exponential to e^(-st) where s is an arbitrary complex number. So this means that if an electrical element has impedance Z(s), then if you apply e^(-st) current, then you get Z(s)e^(-st) voltage.

For example say we chose s = a - bi, then e^(-st) = e^-a e^(bi), an oscillating wave at frequency b that is exponentially damped with parameter a. Say that at that particular complex frequency, Z(s) = Ae^(i theta). Then if we input the current e^(-st) we get voltage Z(s)e^(-st) = Ae^(i theta) e^-a e^(bi) out. In other words, the electrical element amplifies the current-to-voltage by a factor of A, and it rotates the phase by an angle theta. Or in simpler terms, if you are comfortable with complex numbers: an impedance Z multiplies the current by Z to get the voltage, or V=ZI.

Of course in general the current is a superposition of many damped waves, and the voltage will be a superposition of many damped waves too. But each of the single frequency damped waves at complex frequency s in the current will be multiplied by Z(s) to get the voltage.

Now lets apply this knowledge to a practical situation. Suppose we have a simple circuit with a single inductor L. Then we have Z=sL (the impedance of an inductor is sL). So we have V = sLI. Lets see what happens if we apply an oscillating input signal with frequency omega: s = i omega. Then we have V = i omega L I. In other words, the voltage is the current rotated in phase by omega L.

The nice thing about Laplace transforms is that we can not only analyze pure sine waves easily, we can also analyze decaying input, say I(t) = e^(-2t), here s=2 What voltage do we get? V = sLI = 2LI. That is, the voltage is 2L times the current. So we can see that the faster the current decays, the higher the voltage will be (for example if we take s=3).

In summary: instead of thinking of a signal as a sum of sine waves, think of it as a sum of damped sine waves e^(-st) where s is a complex number. Then the impedance Z(s) gives a relation between the voltage and the current for a particular damped sine wave at complex frequency s: V=ZI. So what an electrical element with impedance Z(s) is "doing": it first takes the current apart as a sum of damped sine waves of frequency s. Then it transforms each of these sine waves by multiplying them by Z(s). Then it sums all the transformed waves back together to get the voltage. Of course you can do the inverse process to get the current from a voltage by dividing every wave by Z(s) instead of multiplying. To see what multiplying by Z does you need basic complex arithmetic: if Z=Ae^itheta, then it scales by A and rotates the phase by theta.

If you understand the Laplace transform you understand impedance. If you understand impedance you understand the Laplace transform.