So I almost can't believe that someone can remember how to play, what is it, 36 major triads (allowing for inversions) - oh, no, it's more than that, 72? - and recognise what they look like on the page! Maybe it is just practice and I've only come back to reading music for the last two months or so. I still find myself saying "grass" or "cows"!
Well, sure. There are certainly any number of composers, theorists, and indeed players who probably conceive of things in a combinatorial sense.
I don't suppose the average pianist thinks of various inversions of triads, in, say, E major, as necessarily distinct. I certainly don't, except when trying to deliberately use some forms for a particular sound, usually in a cadence or when modulating.
About your last sentence, if you're not already aware, at least one famous theorist put words (for what purpose, I'm not sure) to the subjects of the fugues from Bach's WTC (I&II). Maybe as a mnemonic, or just a divertissement, but it's a valid method. I know I use some odd visualizations to recall voicings or patterns in octatonic (W-H diminished) scales, which aren't based on anything but, perhaps, necessity.
One thing that I've not seen in all the instructions on learning scales, bizarrely, is whether you should be reading them from a score - my dim memory of 45 years ago suggests that I didn't do this when I was with a teacher, just watched my fingers...or then stared into space once I was better at them.
I wouldn't know what's the "best," but on stage or, these days, just plinking around at my place, I don't really think about it that much. Except in live performance I've learned not to bizarrely gaze at the guitarist's fingers: yeah, I can play a bit of guitar, but I can't really easily decipher which chords by looking quickly at the hands. Just kind of a distraction.
Well, maybe a few, but I'm still unconvinced that this is really all that realistic. Short scalar passages are more often the norm, in my experience, with larger interval jumps, chords, etc., all over the place, and since these often start in odd places, I would have thought it is usual for the fingering to be completely different from how it is if you set off to do a scale (even when you practise a scale from somewhere other than the root note, because, IIRC, you're still supposed to play the "correct" fingering for that part of the scale).
Yeah, I think that's right: it's not realistic, and even in the C maj Invention of Bach, no, you won't be using the same fingerings for every line.
But, IMHO, that's part of the point of, if one wants to call it such, the exercise of being facile with developing fingerings that diverge from the "standard" (and, as you know, even that is not settled).
Here's just a simple example of something I was into maybe a year or so ago: taking bebop jazz tunes and playing the "head arrangements" (really, just the melodies and some harmonies) divided up between LH and RH.
So, take Bud Powell's "Parisian Thoroughfare," in F. It's a very simple, but quite fast melody, at least in the A sections. But, put it in the LH and see how fast you can do it? I can't visualize what I do for the LH, but it's certainly not much like the regular F major scale.
Or, all the other "rhythm-changes-adjacent" tunes in Bb, some of which are, by idiom, played in octaves split between the hands (as one chooses, it's just a popular sound).
No, one must develop a way of playing these things, but while I don't have any proof, having a baseline familiarity with key and at least one way of adapting the hand or the Griff to it, is nothing but an advantage. A significant one, probably.