td;dr: Pickups are over-rated.
Hmm, I'm sure I did this rant somewhere before...
As you already say, many factors contribute to a guitar's sound. And from the start, the sound is all mechanical vibration, so obviously the body etc. can't be neglectable. Clearly, the mechanical parts matter more in acoustic guitars than in electric ones: there, what you hear actually has to pass through the wood.
In solid-body electrics, the impact of the wood is deliberately minimised: a heavy solid body just won't move much to the string vibrations, therefore not much energy is dissipated from the strings, and since the pickups are right there you'd think you get always pretty much "the pure string sound". But the mechanical parts are still significant: with an actual solid rock as body and neck, the sustain would be much longer than in any guitar. The way in which the notes decay and which harmonics stay longer than others is really important, more on that later.
Then there are the pickups. Each of them doesn't pick up all the string's sound, because the partials have different "shapes" as standing waves on the string. For instance, a pickup underneath the 24th fret (if there was one) cannot pick up the e' partial on the E string, the a' partial on the A string etc. This has a substantial influence on the sound, and interestingly on each string in a different way.
The pickups themselves are ridiculously simple: a bit of wire, wound around some magnets. Since it's quite a lot of windings, the coil has a rather high inductance. That single value does almost completely suffice to classify a pickup's sound! Why it is important: an inductor, combined with any capacitor (such as the one at the tone-pot, or just the guitar cable!), forms an LC resonant circuit. Together with some inevitable resistor damping (the volume pot, the amp input...), you get an RLC 2nd-order resonant low pass filter. That is a very simple but quite effective "equaliser" circuit, very familiar to all synthesiser players. Guitarist mostly know in a form with very high resonance: the wah-wah pedal! As we all know, this greatly (or horribly) affects the sound – that's why the pickups have, sure enough, a strong influence on the sound. But really, that's pretty much all there is to them. You could in principle just use always pickups with very low inductivity (that makes the filter ultrasonic, i.e. it sounds completely neutral) and simply plug the guitar into a synthesiser's state variable filter before anything else. Then you could at any time switch between a fat P90 sound and a thin telecaster-pickup (and you'd notice that this still doesn't make a Tele out of a Les Paul, because the bodies vibrate too differently!).
For humbuckers, there is in fact one more point: these use two points of the string simultaneously, so the standing-waves-stuff works out a bit differently. But that's not such a strong effect as most people think: the "fatter" sound of humbuckers is mostly because they have a higher inductance, yielding a lower resonance frequency.
Then of course there is at least the amp, possibly some pedals before it, which do mainly two things1:
Shape / equalise the frequency response. This happens mostly through filter circuits akin to (but almost always more sophisticated than) the LP12 filter that's called pickup. All of this can emphasize or attenuate certain harmonic partials or noise components, but it can't really change what frequencies there are to begin with, or how they change in time.
Distort the signal. That has multiple consequences: you can get "new" harmonic contents (perhaps better described: clone some frequency components into their harmonics), you get intermodulation effects (the characteristic power chord growl, or the undefined mud-sound when you play full chords through heavy distortion), and the dynamic range is compressed. The stronger these effects are, the less you hear of the guitar's original sound
– however, it still matters a lot what you give in: if some harmonics are pushed particularly strong into the distortion, those components will be spread particularly strong through the frequency spectrum, making them much more audible. That's why pickups seem to be particularly important for distorted sounds: their simple single resonance frequency will often dominate where the distortion kicks in. But again, the pickup doesn't produce these partials, it only boosts them. If a guitar is mechanically not able to support a steady sound in the range of, say, 4 kHz, then a pickup where that's the resonance peak will result in an unpleasant, short clang from the distortion, while a good Strat would be able to make a cool screaming lead tone of this. If you put the resonance lower (perhaps 2 kHz), the same Strat might sound rather dull, because those partials simply decay slowly and don't sound very interesting by themselves. OTOH, that's the very range where Les Pauls unfold their characteristic sweet singing tone, but that originates from the body and neck.
This is probably what Seymour Duncan are aiming at with their recommendations for various wood types. It's certainly the reason why these pickup/body combinations are most popular. But definitely there isn't one right pickup for each guitar. I'd rather see the pickup as a powerful but crude selector for what you want to center your sound around. Replacing the pickups isn't a very effective way of doing this, if you want to switch this between songs or even during one. Since it's only passive filtering in any case, you can always approximate it quite well with extra EQing before the distortion. Or you can do a trick: changing the inductance is, resonance-freq-wise, equivalent to changing the capacitance. Normally, the capacitance is main split up in two parts:
the guitar cable. Urgh! Who wants that to have a strong influence on the sound?
the tone pot capacitor. That part is almost disconnected when the tone pot is fully up, then "attaches" the circuit when you close it. Unfortunately, in between there is the resistance of the tone pot itself, which damps the resonance, so this capacitor can't really be uses to control the boost frequency.
A solution to this is an active circuitry right in the guitar. Strangely, guitarist seem to have some kind of allergy against this idea. One reason is probably that the most prominent manufacturer of active pickups, EMG, has famously taken it to the extreme: the EMG81 has a frequency responce you can't reproduce with passive pickups. Most other manufacturers do the opposite, just leave the resonance where it is "traditionally" (but in a way that it won't change with the outside capacitance, which isn't all that great either).
1I shan't talk about more complicated effect pedals here – of course you can go to arbitrary length with those; at some point you will only hear the effects and not the guitar at all...
