In summary, I guess what I'm asking is, am I missing something, or do I just need to go shopping for a better instrument and better equipment? I mean, how good can I make an entry-level bass with less-than-average pickups sound?
In this case, I'd say no, you're not nuts and the bass is very likely a limitation for what you want to do. Besides that, you're not the first one who's had trouble getting a decent sound out of a Fender Jazz bass anyway - you really have to be an athlete to get them to really respond to start with, no matter how good you are (i.e. Jaco or Geddy Lee, both of whose arms and hands are stronger than a mountain gorilla's).
So I agree that you're already in a bit of an uphill battle with the bass you have. As for how much further you have to go in terms of cost and quality; that's pretty much an individual thing that depends on your current skill level and economic leverage, etc.
What you may find is there's a particular level of quality and cost above which any difference starts to level off. I.e. in my case, it's about $1300 - any more than that and I sound and play the same. That's good, because I'm not richer than about $1300 anyway

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But on my Carvin BB70F, I play and sound as good as I'll ever be able to play and sound, no matter how much more expensive the bass is. You could put a Fodera Anthony Jackson in my hands for over 20 large, and I won't sound or play any better than on my Bunny 6 string. And in fact I may sound worse for fit reasons.
Again, that's an individual thing for you and probably something you'll have to research for yourself. Try various different instruments and see which ones fit you the best and sound the way you want. Also, be mindful of the cost, how many of your kids will you have to sell to a South Pacific drug lord to buy the thing, etc. All that will go into the equation...
my .02,
LS