These discussions usually seem to arrive at a middle point.
Yes, wood almost certainly plays some part in the overall sound/tone of a solid body bass, but no one has conduced suitably rigorous tests to confirm how much, or little that part might be.
The specific cases often referred to, where players describe a noticeable difference between bass A and Bass B, may well be quite right, but wood iss a variable natural thing and the next two basses compared with
I think players try to dictate what makers do too much. If I was in the market for another instrument my instruction to the maker would be much more limited than earlier in life.
I am not a luthier but if a client was asking for an instrument that didn't meet his criteria, I would tell him that his request would likely not meet his expectation.
If I thought the instrument was going to sound bad or look bad, I wouldn't make it. I can't see how making a bad instrument with your name on it could be good. If the customer couldn't understand that I was trying to help him, then I wouldn't want his business.
If the instrument was otherwise going to be good, I would make it. I would accept it back if it didn't meet his expectations during the 1 week trial and sell it to someone else.
I think you would make a good customer. (Definitely not touting for business, I'm pretty much retired and probably on the other side of the world from you...

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Coming from the other side, as a maker, I never took a commission that I thought would be a disaster in terms of the instrument as specified, not meeting the customers stated needs, nor would I take one from a customer who keeps changing their mind, but that's a whole other story.
I certainly wouldn't want my name on an instrument, I was pretty sure the customer wouldn't be happy with, that is a sure fire way to get a bad reputation. Because very few customers are going to tell their friends that they are the cause of their unhappiness with their shiny new custom instrument, because they insisted on bad choices over what went into the instrument. No, they're going to blame the builder for their disappointment.
Once reaching an understanding with a customer, it's important to stand behind your product, but I'm afraid that in most cases with a custom instrument, especially if it' incorporates some unusual specs, (extra wide neck, very radical shape, unusual finishes, unusual scale length, etc.) may well be near unsellable to anyone else other than the person who ordered it, or certainly will only sell by lucky circumstance, or by taking a heavy loss on price. For small scale one person luthier operations, the hundred or so hours wasted, value of woods, finishes, frets, etc. used can be a huge blow.
I don't think the wood makes a bigger difference to the TONE than the pickup if we're talking about two completely different types of pickups. But for things like dead spots, sustain and having an even response along the neck then yeah. For those things the wood definitely matters more. In fact the pickup probably means nothing for these things (although I won't completely rule out the possibility that it might have a marginal effect due to things like differences in magnetic drag, etc.). People tend to only think about TONE (as in the frequencies we hear) when we're talking about tonewoods but the way the instrument responds dynamically is perhaps even more important IMO. Wood has a HUGE impact on these things.
If we're comparing two pickups of a similar type (like two "vintage voiced" P-bass pickups from two different manufacturers) I actually think the wood can make a bigger difference, even when it comes to TONE. I don't just mean the TYPE of wood but the QUALITY of the wood too. I think I clarified this in an earlier post.
The POSITION of the pickup (which is a huge factor when it comes to your example of mudbuckers) will certainly have a bigger impact on tone than both the wood and the pickup itself (I think I also mentioned this in an earlier post). In fact, the term "mudbucker" is often used for the position of the pickup as well as the pickup itself.
I get the impression that lots of people don't realize that tonewoods isn't just about TONE but also about dynamic RESPONSE. Even the most expensive boutique pickups can't fix an instrument that has a poor response, poor sustain, severe dead spots or an uneven response along the neck. These things are usually the biggest difference between a cheap instrument and an expensive instrument these days, not the "tone". And these things are most definitely affected by the mass/density of the body/neck. In other words: The wood.
Half the "new bass" posts here on talkbass seem to make the impossible claim that the new bass doesn't have any dead spots so I guess lots of people aren't really aware of these things.
I agree that the quality of and the strength, degree of flex, or rigidity of wood are viable and often positive features that can be different and measurable in different species of wood and even different pieces of the same species. But that isn't what most people are talking about when they refer to 'tone woods', or ask about how much of an effect on the tone of a solid bodied bass comes from the type of wood.
Certainly some wood species offer greater rigidity, density and are less prone to excessive shrinkage, or warping, twisting, spliting, that others. All of those things can make them more suitable that other species for necks in particular.
I started out in my life with wood learning cabinetry, that teaches you a lot about the structural qualities of woods, I also studied sculpture, specializing in wood as a medium, which makes you think about the grain and stresses within the wood, important if releasing those inherent stresses aren't going to lead to a finished work spliting over time.
I believe good luthiers understand wood and many of the decisions about which woods became "standards" for instrument construction was at least as much about the structural qualities of the species chosen , as about the woods, tonal qualities. It doesn't matter how good a particular type of wood might sound, (though any evaluation of that aspect is a personal one anyway) if the neck is unstable, the fingerboard to soft, or the body splits apart easily.
Could it be that the "tone woods" we have been conditioned to, are actually structurally suitable woods for making lasting stable instruments and we've learned over hundreds of years to accept the tones they produce, as the "best" tones?
Whatever the truth might be, we have seen a huge expansion of the woods used and proclaimed acceptable as "tone woods", over the last few decades. The days of the likes of mahogany, rosewood, ebony, maple, ash and alder, ruling the tone wood roost, have morphed into those woods being just a part of a much wider palate of 'suitable' woods.
None of that answers the old "how much of the tone of a solid body electric bass/guitar, is the wood responsible for?" Though I think the end answer has to be something like, 'it's variable, ranging from a little to a clearly noticeable amount, defendant on the individual pieces of wood used in the instrument as they interact with all the other components and are hear in a particular circumstance. But can we accept that not every piece of ash, alder, or whatever, will sound exactly the same as every other piece of ash, alder, or whatever?