Is the only advices that are needed to play chromatic scales with 1212 and playing scales & arpeggios?
I wasn't really following this thread, so I missed this suggestion. Intriguing, but I've never tried it (I just use the "regular" fingerings for chromatic scales, i.e., from C, 1 3 1 3 1 2 3 1.....etc), so can't comment other than that it sounds like it's worth a shot.
What I have been playing with is the Chopin étude Op. 10 no. 2, and the corresponding Godowsky version for LH alone.
While I have not been very diligent nor successful with the peculiar fingering Chopin uses (and which Godowsky sort of inverts for the LH), the thumb does come into play in the accompanying chordal figurations.
So, regardless of the rest of the RH, literally all the thumb and index finger are doing is dropping very quickly onto the keys. Again and again. I don't see how you could possibly play it with any tension at all in those two fingers. They just fall onto the keys, since fingers 3, 4, 5 are very much carrying the piece.
I would think an even better exercise, with an equally slow, attentive pace required, would be just doing repeated notes using 1 and 2. Or trills using the thumb on one of the notes. Either hand.
I doubt you'd be going at too fast a pace with those, but you can observe (or have someone else observe) what motions or tensions are contributing to your difficulties.
But, in generalities, I can't say more than what others have suggested: don't try to move your thumb at all. Attack trills (including single-note trills) using only the wrist, and see if your thumb doesn't adapt to this "forced relaxation."