I'm going to say that on the whole it doesn't, because pickups don't work that way.
If you mic'ed a few electrics, the density of the wood and the physics involved may allow for some differences, but pickups themselves don't work by the same processes as a mic does.
Then once you add in a warm amp, some cabling between amp and guitar, possibly between head and cabinet, the EQ settings and all that.
Then the drummer,
Then the bassist,
Nuance in the guitar itself is lost, now you are using the tone shaping properties of the amp's circuitry.
You point about the guitar's sound being just one factor in an often complex chain and mix is certainly valid. But the same could be said about a Stradivarius in a full orchestral recording. Would I notice the difference if it were played by the 2nd violinist? Probably not. But by the soloist in, say, the Brahms Violin Concerto on a DG close miked recording? I suspect yes.
Also there is that factor of the strings vibration themselves being influenced by the rest of the instrument. They are -see the discussion in the "Hot Or Not? Ibanez ASR70" thread. No instrument is entirely inert. Nor, in most cases, do we wish them to be. Overtones are created when certain frequencies excite the non pickup, non string, portions of the instrument and vice versa where certain frequences fail to excite or even damp those frequencies.
That is why nut material matter. And bridge material and design.
Eric Clapton, for instance, would not approve a Strat to bear his name with that had either a live floating vibrato bar or one with a fixed stop piece. The sound that he favored could only be created with a blocked off floating vibrato..
The point is not that a guitar can only sound good set up that way but that such a change made a difference that was important to him.
Some Tele players favor one bridge design over another -- and the combination of brass and steel seems important to many. Same with the maple fingerboard vs one made of rosewood or rosewood vs ebony on a Gibson instrument.
Yes, if you load up your chain with enough pedals or over drive the tubes or diode circuits those subtleties can be lost, just as adding a lot of salt can hide the carefully blended and balanced flavors of a good marinara sauce. But that doesn't mean the such careful choice of ingredients and spice blending itself never matters.
-don