It might be worth our time to consider a few of the basics of troubleshooting methodology. In reading the thread, I'm seeing the following things considered(in no particular order):
1. Type of body wood used
2. Neck wood(quartersawn vs. flat)
3. The combination of pickups from different manufacturers
4. Weak battery
5. Wiring issues(pickups out of phase, mis-wired capacitor, etc.)
6. Component out of spec.(Resistor, etc not within specified tolerance)
7. EQ at the amp
8. The type of bridge used, and it's transmission of audio information to the body
9. Use of possibly-inappropriate values for pots
10. Bad ground connection
11. Use of specific components(Stellartone, pickups, etc.) lending this specific character to the tone(with the assumption that no anomalies exist).
12. Pickup to string distance
13. Type of string used(already addressed by changing strings)
Along the way, we have chosen to troubleshoot the tone on a different instrument(If I understand correctly that the example in the video is NOT the bass in question).
A quick look at the list above will indicate which of these items can be tested with relative ease, as opposed to more complicated changes. It is my suggestion that you eliminate the possible causes in that order, by taking the "low-hanging fruit" first, then advancing to more complicated tests.
If you have not done so, I would strongly suggest(because I, like others, believe the answer lies in the electronics) that you photograph the control cavity, then use the photo to help you draw a schematic diagram of the electronics(if you have not already done so). This could be very helpful in determining the cause of your woe, and some measure of documentation would be very useful to anyone on the thread who is trying to help. I'd also suggest limiting the conversation to your bass only, as other similar-sounding examples may sound that way for reasons that are completely different from the conditions surrounding your build.
Best of luck in getting this the way you want it to sound. I'll continue to watch, as I am curious to learn the eventual answer.