Hello again, all. First, please let me reiterate... I am an idiot. I am a giant, flaming bag of poo idiot.
As mentioned, the inside of the bass was a mess, which I would have likely noticed if I had paid attention to the cavity when last I was in there... which, admittedly, was several years ago, and I have learned lots since then... But, still, I am an idiot.
In comparing the guts to the many, many online diagrams available, I discovered many irregularities, which I set right. (the odd wire mentioned above was actually for the tip of the output jack...) And...
Holy Flurking Schnit! It is loud, punchy, clicky, meaty, reedy and insistent... all of that. It sounds like a Jazz bass now.
I threw some electrical tape over the poles, and all is now right and good with the world. You may all rest easy.
I may, possibly, still part with this bass someday, or at least upgrade the PUs, but, for now, I'm gonna attempt to learn this Jazz bass with it's various oddball quirks.
I do wish it was as easy as a P bass...
My seriously sincere thanks to all who chimed in here- all of your advice led me to a solution (for now), and, for that, I am very appreciative. You all deserve a cookie. Or several cookies. I'll leave that up to you. Basically, all of you should go get some cookies.
Thank you.