I think ultimately your original answer is flawed in its premise because I actually do understand the notion of a “true” humbucker as in something that resembles a PAF as Seth Lover designed it, a large pickup with coils in series (which actually excludes the original MusicMan humbucker, funny enough). And plenty of basses do have them. Most soap bars are internally exactly that and it’s quite easy to get that tonality as a result.
Where your premise ultimately is flawed is that even in the guitar world, mini side by side rails sound nothing like PAFs. The actual size of the pickup sensing area is a huge part of tone. This is why a PAF sized P90 never sounds exactly like a P90, even when you use the same design, materials, and windings. At the risk of grossly simplifying, usually overwound coils will sound stronger in the mids and putting coils in series will give you bolder mids and higher output. To get a truly warmer and bigger bass, I have found that you often need physically bigger pickups. You can’t just wind your hot rails hotter and expect it to sound like a PAF. This is why MM pickups, despite the scooper clarity of their parallel wiring, still have some of that huge sound. I would expect split coil P90s to sound warmer than Strat sized hot rails. In fact, side by side rails are one of the many designs used in the guitar world to get single coil sounding humbuckers. GFS Neovin, Lace Holy Grail, and MojoTone Quiet Coil are side by side humbuckers (some with purely cosmetic pole pieces) that are advertised as sounding exactly like a single coil. Meanwhile, to get a PAF sized pickup to actually sound like a single coil seems very difficult unless you just physically use half the size of the pickup.
Anyway, the point I’m making is that for whatever reason, Jazz Bass pickups are rarely made in rail form. And because of that, you might assume that somehow the splits are not giving you what the rails promise. Meanwhile, in the Strat world, split coils are practically unheard of. In both worlds, stacks are often frowned upon for their supposed lack of clarity, though they’re much more researched and accepted in the Strat world. But they usually sound much more like a single coil than they do a PAF. And I believe this is due to size.
Not all Jazz splits are designed to sound like a Jazz Bass single, though. For instance, DiMarzio makes three: Area, Ultra, Model. Only the Area is supposed to sound like a single. Ultra is a high output, modern pickup that can match the output of a P neck pickup but still sounds like a J to some extent. Model is a beefy, medium output pickup that really doesn’t sound that much like a J. Most of the boutiques do try to basically sound like a J or a beefy J, but any of the custom shops could fix that up for you. The Duncan Apollo pickups, which are not boutique but they are what I use in my PJ basses, they sound a little bit warmer and bigger than a real J as well. Some people consider that a problem, I consider it a feature.