This is very true.
The physical characteristics of the frame between the anchor points of the strings plays a big role too. Stiffness, density, etc.
Just like strings would vibrate/sound different if you anchored them to either end of a steel pole, than they would if you anchored them to either end of a willow pole of the same dimensions. It's the physical characteristics of the frame and not so much simply due to the species or color of the wood.
I think that's where the confusion lies with what types of wood affect "tone". There's no inherent sound ascribed to any species.
The finished product and how it's constructed is what determines how the strings behave and what the pickup tranduces.