Can you bend your thumb nail joint inward at all, from any position? From your vid I can't help thinking it's psychosomatic. From the pic it seems your only problem is not being able to bend the top joint - joints below look fine.

Personally, I'd say the very opposite. Why does the top joint need to bend inwards anyway? While it's worth trying to improve range of movement, there's no need to bend that joint much-and especially not to try to brace it. In that piece, I'd only bend it very slightly, before straightening out again during the depression of each key.
What I'd be concerned by is the rigidly braced joint lower down. It's severely cramped up. Instead of bracing it, I'd be extending the thumb out to its full length- to put an end to that cramped position. I have a student with a very similar thumb who was similarly stiff in it. The action of extension has resulted in a wealth of improvement. His thumb no longer looks anatomically strange when he plays, but now appears perfectly normal. Particularly in the case of slightly unusual anatomy, extension of the thumb is the simplest way to use it. There are various exercises for this in the second post down on my blog. If the top joint will not bend, just concentrate on getting the others moving, rather than locked stiffly.