The "problems" with "stock" Jags and JMs are due to unsuitable setup. They originally came out of the factory set up fine. They had neck shims and flat wound 12s on them, and were made to be played "clean," for old style music. Put round wound 9s or 10s on them with no shim, start playing modern distorted music and rock guitar solos on them, and you'll obviously think they suck. It's like driving a race car with cheap-o, undersized street tires on it. Set them up perfectly, and they are problem free. No expensive new replacement parts (bridges, tension bars, etc.) required. Problem is that most players, and even techs, don't know how to set them up very well.
The original wiring circuit works great IME. You just have to become adept at using it, and approach it differently than a Strat or a Tele. JMs were designed to offer a huge tonal palette from dark to bright, while Teles and Strats occupied a more narrow band. Having a wider tonal palette means that you need to take a more decisive and active role in telling the guitar what it is you want from it.
The rhythm circuit dimed (1M volume and 50K tone) sounds very similar to the neck pickup on a Strat; it isn't just a "dark" circuit (as many think of it) unless you have the tone knob rolled down. The lead circuit on neck pickup sounds like a Strat middle. And the lead circuit treble pickup sounds like Strat bridge. So if you want something closer to what you get from a traditional Strat or Tele, you will find your best use for the guitar to be switching between rhythm circuit dimed and lead circuit on the neck pickup, and setting the amp for those two main tones. NOT using lead circuit neck and lead circuit bridge as your two main tones, and setting the amp for them; doing that leaves your rhythm circuit too dark (the problem most people experience).
Also, you're dealing with 1M pots, which are designed not to be used dimed during normal play – only when maximum treble is desired. The point is to give you a wider tonal palate, which you then manipulate into the range you want, using the onboard controls.
The only real "problems" with these guitars are that people don't know how to set them up, and don't know how to control them. They require you to tell them exactly what you want them to sound like, and they are unrewarding to those who just want to "plug-n-play."
FWIW, I am a Tele and Strat guy too, for the most part. But when it comes to exploring the extremes of controllable tonal range in a vintage style passive instrument, the JMs can't be beat. If I had a recording session and could only have one guitar, it would be a stock vintage style JM, hands down. It can do anything.
Also...how many of you play the stock vintage setup on Strats and Teles? I'd reckon that almost all of you use modern wiring on both, i.e. five way switch and bridge tone control with Strats, and modern switching on a Tele. So don't knock that JMs "need" mods to work better with modern style music...because so did Strats and Teles.