Tonewoods

The development of the electric guitar, probably, along with the electric organ and electric piano the most significant new instruments created in the 20th century, has led to some interesting perspectives on the traditional instruments. For one thing, we now need the description "acoustic guitar" when talking about guitars that are not electric. But today I noticed something quite odd. My intuition and experience tells me this cannot be correct (screenshot from an Instapundit comment section):

Click to enlarge

The claim is that the wood chosen for the body of a solid-body electric guitar is going to have a significant effect on the timbre. Can this possibly be true? Isn't it just a bit of knowledge relating to acoustic guitars spilling over into the electric world? Yes, the exact characteristics of the wood of the soundboard in particular of acoustic guitars is hugely significant in terms of the timbre of the sound. But I don't see how it can have any effect on the sound of an electric guitar. The sound from an acoustic guitar comes from the vibration of the string making the bridge move which in turn makes the whole of the soundboard vibrate in complex ways. This vibration creates compression waves in the air which we hear as sound. But electric guitars function in an entirely different way. Here, from the Wikipedia article, is how they work:

Unlike acoustic guitars, solid-body electric guitars have no vibrating soundboard to amplify string vibration. Instead, solid-body instruments depend on electric pickups and an amplifier (or amp) and speaker. The solid body ensures that the amplified sound reproduces the string vibration alone, thus avoiding the wolf tones and unwanted feedback associated with amplified acoustic guitars. These guitars are generally made of hardwood covered with a hard polymer finish, often polyester or lacquer. 

The degree to which the choice of woods and other materials in the solid-guitar body (3) affects the sonic character of the amplified signal is disputed. Many believe it is highly significant, while others think the difference between woods is subtle. In acoustic and archtop guitars, wood choices more clearly affect tone. 

Compared to an acoustic guitar, which has a hollow body, electric guitars make much less audible sound when their strings are plucked, so electric guitars are normally plugged into a guitar amplifier and speaker. When an electric guitar is played, string movement produces a signal by generating (i.e., inducing) a small electric current in the magnetic pickups, which are magnets wound with coils of very fine wire. The signal passes through the tone and volume circuits to the output jack, and through a cable to an amplifier.[32] The current induced is proportional to such factors as string density and the amount of movement over the pickups.

Here is my thinking on the subject. The extremely tiny effect that the solid body has on the tone is evidenced by the almost completely inaudible sound you get when you pluck the string of an electric guitar that is not plugged into an amplifier. The sound really comes from the signal created by the movement of the metal string in the magnetic field of the pickup. This signal--which has no acoustic component, let it be noted, is then amplified and modified by tone controls on the guitar and the amplifier and is often further modified by a whole panoply of other effects activated by foot pedals. Whatever tiny effect the acoustic properties of the body might have are completely drowned in the signal processing.

How could the acoustic properties of the wood body possibly effect the sound of an electric guitar? I welcome comments on this question, because I can't see how that could work. I'm not of course, talking about hollow body electric guitars which would be a quite different question.