I don’t think convincing is even a goal , but just to add to your thoughts on this.
1. A Precision sounds mostly like a Precision due to the placement of the pickup, not the pickup itself. You will absolutely hear a difference from a Swamp Ash body or Alder body P. You will less difference if you change the P pickup brand, you would hear most difference if you would move the pickup forward or back an inch.
2. A slant Jazz pickup in the P position is hardly distinguisable from a P pickup. Again pickup placement has most impact.
3. We agree that strings have significant impact.
4. Take a Passive bass or Active bass with passive tone knob and roll that off and it will behave roughly the same. Sure an active EQ gives you more control and flexibility but if you have a transparant onboard EQ the passive tone and no EQed active tone will be pretty much identical and an active EQ certainly will not correct a lack of resonance in the wood or a badly selected pickup for that matter.
But to be fair, this is based on my experience, my ears, my experience with bass builds etc.