Back to the two cabs from the OP...I've played both the TC115 and the F112 in a small blues context, and both are great in that, they take what you give them, but I guess I don't understand the concept of genre based cabs.
That sounds WAY snarkier than what I actually mean to say, but my point is... A well designed bass cab, to me should just accurately spit out whatever you feed it. Example, I love Slim Harpo, Howlin' Wolf, Little Walter, etc. Willie Dixon is my all time hero, along with Carol Kaye and Duck Dunn. I pick my favorite basses for the intended sound, but I use the same cabs for dirty blues gigs that I use for punk rock and pop country gigs. Having great cabs allows me to focus on other things when selecting the right tools for the job.
You are correct that any piece of gear can be used for any type of music. However, you are incorrect that any cab 'accurately spits out whatever you feed it'.
ALL cabs have a strong tonality, and no amount of knob twisting can make one cab sound like another. It is not about 'genre specific' at all, it is all about matching a players tone goals and volume needs as easily as possible.
I also use the same cab pretty much for most gigs I do, no matter what the musical genre. However, that cab has a very specific sound that works for me with a minimum of fuss. And I have another cab or two that sounds VERY different from my main cab, and does some things better than my main cab, and sounds better to me at certain volume levels with certain instruments (and yes, with certain styles of playing that seem to benefit for a certain tone).
Put another way, I also have a great cab that allows me to focus on other parts of the input chain and music. But you know what, my great cab sounds WAY different than your great cab. That is the way it works with backline amplification of a 'non programmed' sound source.

