I know you and I have talked about neck construction being a big part of the equation. My Wal-ish bass has an all wenge neck because the basses I've played with them exhibited a more dense mid-range and brighter tone, and I also own four different basses with carbon fiber necks. Most other things being equal, carbon fiber is immediately distinguishable from the rest. I've always felt that the stiffer the neck gets, the more the tone starts to open up and noticeably change. Now whether that changes for the better or for your preference becomes a personal taste. Not everyone likes the sound of a carbon fiber. There's no denying it's a different tone (usually brighter), plays very evenly, and has sustain for days (can't emphasize that enough!). So why then, when it comes to wood necked basses, would that be any different? I've played on plenty that were dead as can be, and others that would just absolutely sing. Wood, is not a (mostly) synthesized and controlled product like carbon fiber, so it inherently is going to exhibit differences. Wal figured it out a long time ago in their process that if they dry it to a certain humidity and do a multi-wood construction that the tone will remain relatively consistent from one bass to the other. I bet if you changed the neck on a Wal to carbon fiber that it would still sound like one, but different characteristics would start popping out and others might disappear.
Body wood remains on the bottom of the list for me for a major tone factor in a solid-body, bolt-on neck construction, with a top loading bridge. I think there is a good relationship between the neck and the body and the connection between the two needs to be tight to resonate properly, but the body wood itself is less of a factor. The bridge I feel also does more than the body wood does, and I've experienced that first hand swapping them out on a few basses and hearing the immediate difference. Wal uses their own design which is not found on anything else in the world, so unless someone swapped their original for something else it would be hard to say how much it is a contributing factor...but I bet it's enough to notice.
I've also had a conversation with a buddy of mine who has owned hundreds of guitars and works at a boutique guitar company. On a strat style setup the pickups are mounted to a plastic pickguard, and if equipped with a tremelo the bridge floats between the two posts and the springs. So what tone are you actually hearing in this case? The neck? The pickups? This could bring the argument that a lot of bass pickups are screwed to the body wood and it might make a bigger difference, but then there's Wal which functions more like a Gibson style humbucker that floats on a pickup ring. So again, what tone combination are we actually hearing? In my mind, and minus the type of strings, it goes like this: Pickups>Preamp>Neck>Bridge>Body