I think the more you amplify it, the less it matters. Wood properties mostly have to do with resonance - density, weight, etc. Resonance has very little impact on what the pickups pick up. There’s some harmonic content, but most of it is drowned out in the amplification process. Acoustic instruments are obviously much more influenced by material and resonance. The sound you hear with the guitar unplugged isn’t directly translated to the pickups.
I’d say a rough list of what influences tone would go something like:
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Play style - ie Van Halen sounds like Van Halen regardless of what he’s playing
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Set up - this will influence how you play (fretboard material might come into play here), as well as stuff like string action and pickup distance from the strings, intonation, etc
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Amp - an AC30 will never sound like a JCM, no matter how hard you try
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Rest of the signal chain - pedals, etc
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Pickups - kinda far down the list, but can have a decent influence on overall tone
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Materials - subtle, sometimes imperceptible differences
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After years and years of playing lots of different guitars, I really think stuff like Alder vs Ash is more a psychosomatic thing with the player than a measurable difference once you apply the sort of amplification that modern guitarists use. Amps have a relatively narrow frequency bandwidth with massive high/low rolloff from the pickup impedance and preamp, so whatever shimmery highs and booming lows your wenge thru neck might have is going to get lost when you plug it in.
tl;dr - yes, materials can play a part in tone, but it’s a very, very small part that usually isn’t worth getting hung up on.