TIJFs are supposed to be the holy grail of bass strings, but I've read enough about them to know they're not for me personally. Just a personal choice...
TI flats are rather unique strings so you tend to get very exaggerated reviews from both the people who love them and the people who hate them. I don't think you will ever truly know whether these strings can work for you unless you actually try them yourself.
They're definitely not some kind of magical solution to everyone's strings problems and they certainly won't be suitable for all playing styles and all genres of music. Saying they're the "best" strings ever (like some die hard TIJF fans tend to do) is quite frankly ridiculous. This will always depend on the playing style and the style of music.
On the other hand, their floppiness is a bit exaggerated IMO. Personally I can't play 40-100 roundwounds. That's too floppy for my playing style, I get terrible fret buzz even with a perfect setup. But I can easily play TI flats with normal action. I suspect the internal silk layer may dampen the vibrations somewhat, which perhaps make them less prone to fret buzz than the gauges and supple feel may suggest. The silk layer doesn't seem to affect the sustain much though. They have far better sustain than any other flats I've played (and I've played most). Ideally I would have wanted the A-string to be SLIGHTLY stiffer but the rest of the set isn't really that floppy IMO. At least if we're talking about the long scale set. The short scale set isn't nearly as nice IMO.
If I had to describe them I would say:
-By far the most articulate flats you'll ever find. In fact, they're one of the most articulate strings you'll ever find in ANY category. The A-string is a bit light tension for my taste but nothing I can't handle (as a reference I prefer the stiffness I get from 45-105 with most roundwounds).
-Brighter than most flats but not in a zingy/metallic way (except maybe for the first 5 minutes). It's more about a nice, smooth upper mid presence. A very "organic" tone.
-They are known to "break in" really fast but in truth they keep mellowing out over time. When comparing sets that are 1-2 years old with sets that are 10+ years old they sound VERY different. Both sound great but the older ones are significantly more mellow. Still VERY nice though. Perhaps even nicer.
-The lows may seem a bit lacking when comparing directly with other strings but in reality they are VERY nicely balanced. After playing TI flats for a while I actually find many other strings to have bloated lows. It will depend on the bass/amp/eq settings of course.
-They seem to last practically forever when installed on a bass but they do NOT seem to hold up well if the tension is completely loosened (for maintenance or moving them to a different bass). I've ruined several sets this way.
-Definitely not the right strings for an "old school" flatwounds tone. I see them as a more articulate and smoother, more "organic" sounding alternative to roundwounds.
I'm not saying these strings will necessarily work for you of course. I can't possibly know that. But I honestly don't think you'll ever know without actually trying them. The gauges, and the supple feel of the TIJFs shouldn't work for my playing style at all, in theory. But they do. And the tone, on the right bass, is absolutely gorgeous! I particularly love them on Danos and single coil P-basses, where the nicely balanced lows and the lovely, smooth upper mid presence balance out the inherent scooped/boomy character of these instruments. I really like them on J-basses too. With many split coil P-basses I find them a bit too mid focused for my playing style. It will depend on the pickup of course. With darker sounding split coils, like the PV63, I like them a lot.
Do I use TIJFs for every song? Absolutely not, but I probably could if I had to.
Can I ever see myself NOT owning a bass with TIJFs installed? Absolutely not.
The TIJFs aren't for everyone of course. If I only owned one bass I would probably not use them. But they are totally unique and every serious bass player should try them at some point IMO. You REALLY won't know until you try them yourself. And if you ever do, allow yourself some time to adapt to their unique "feel" before you make up your mind.