So, to be clear, you are essentially asking whether pickup strength affects the degree of influence that the mechanical construction of the bass contributes to the individual instrument tone? And / or speculating that more powerful pickups over-ride that factor?
Probably, the best example to look at would be the Alembic Series I/II instruments. Quite aside from their eye candy construction, Alembic has an ultra powerful pickup drivetrain; and the neck through construction is specifically used, in their case, to minimize variation in tone between instruments. The engineering design intent was to use the woods for eye candy and NOT for tone. On this topic, you might have an interesting discussion with Ron Wickersham.
Yet, for all that effort, no two Alembic basses sound or play quite the same. And part of, or perhaps the main problem for us players is that it is generally not possible to separate the player interaction with the instrument from the sound the audience or recording board hears, even though they are two separate aspects.
My dear wife will tell me the basses all sound the same. And, indeed, it is quite difficult, perhaps not possible, to differentiate two different basses of similar configuration on a recording. Which is one very good reason to remain focused on playing skills and knowledge.
Still, there is a very definite hookup that occurs between the player and the instrument while playing that quite definitely includes the mechanical aspects of the individual instrument construction, and strongly affects the act of expression. In short, we perceive a whole world of somewhat subtle aspects of the instrument that are not primarily apparent in the audible product. I suppose that's why we keep going around in circles with this conversation.
Anyhow, you've got a vibrating string that carries an astoundingly complex array of overtones; which themselves are subject to the specific damping characteristics at the string ends. The pickup converts the final summation of that complexity. The only way to take wood out of the equation entirely is not to use it. Instead, you can use a graphite composite or aluminum; and instruments with those kinds of construction can be produced with impressive instrument to instrument uniformity. Is that what we really want as players? The market has generally said, no.
Going back to Alembic, my experience is that their philosophy works to a degree, and that supports your hypothesis. But, ultimately as a player, the mechanical construction of each wooden instrument will matter.
And, we will probably keep debating it.
In fact, Alembic puts a brass block underneath the bridge, explicitly to decouple/isolate the bridge from the body.
They intend it to be an inertial mass that reflects energy back into the string rather than absorbing it.
Ron Wickersham has been clear about its intended purpose - read the BP article from the early 2000s and he talks about it.
This, to me, further supports the case that wood matters. If you have to design a system to minimize the impact of the wood,
and if the material underneath the bridge makes a meaningful difference, then both observations point to the same conclusion - the material is an important determinant of sound.
Alembic also describes the tonal effects of the different neck laminates on their website.
Most of the arguments suggesting wood doesn't matter could be equally applied to any other mechanical/physical component of the bass.
Sure, if you roll your tone to zero or boost the active bass control to 10, then you might not hear the differences between a bent-plate Fender bridge and a Badass.
But that doesn't mean they sound the same.
Ditto for the YouTube "science", which is never really well-structured IMO. Pressing a neck/bridge structure into some blobs of wood will not prove anything,
especially once you add in the compression YouTube uses to process audio.
To OP's question - certain pickups can have a nonlinear response that is distinctive enough to mask or filter out some of the effects of construction.
EMGs, for example, have a particular tone that I hear coming through no matter what I put them on.
Gibson "Mudbuckers" are another example.
Remember, as always, that these simply represent a "filtered" reproduction of the instrument.
If you look at frequency response analyses for common bass pickup types (J, MM, etc.) you'll see that they have substantial peaks and dips, and are far from "flat".
In my repair work, I fix a lot of broken Gibson headpieces. Often, in bad cases, I have to laminate the back of the headpiece with ebony to strengthen the cracked area.
Doing so makes the headpiece WAY more rigid, since it has "skins" of hardwood on both front and back.
It is very common for those clients to report that their guitars sound better, clearer, sustain longer, etc. after such a repair.
This is because the headpiece becomes less of an energy sink. I don't solicit these observations - people just tell me afterward that it's better.
Another clear example of the effect of construction.
Cheers,
Martin