It's physically impossible for me to do what he is doing with the "quiet hand". Beyond the first couple of small movements - I simply do not have a thumb that is long enough to reach the pass under the 4th. No amount of practise or exercise will change that and although he angles his hands slightly, the amount I would need to would be ridiculous, I wouldn't be able to cover the other notes.
Dear Leahcim,
Throughout my life I had maybe about 200-300 students (starting from age as young as 4) and don't remember one single case when the thumb was too short to do the task.
It is very hard to give any advice without seeing what you are doing, but it is clear that you do something not quite right.
I will try to outline at least where to look.
The most common mistake is treating thumb as a finger with two joints. In fact it has three, where the third located close to your wrist.
Put your hand down, feel it absolutely free, and start slight vibration of your arm. You will feel your WHOLE thumb being completely relaxed, and you should get the feeling where it starts. That's how you should feel it while playing AT ALL TIMES--straight, flexible, and relaxed, with its motion starting from the 3rd joint.
Now, take your completely relaxed R.H. and try to slightly squize it with your left hand fingers placed on third joint of the thumb and third joint of the pinkie. You will feel that the palm gets a slight tube-like shape. That's what you should feel when the thumb reaches under.
Now make an excersise. Put your completely relaxed hand on the keys with thumb on the F. Press it gently, putting only as much weight, as it is nessessary to hold the key down. Feel your hand relaxed. Hold the F with your thumb and lightly press G with your 2nd finger. Release the 2nd and slowly and smoothly (while still holding the thumb) move your 3rd down on E. The thumb should feel straight, relaxed, and "growing" from the 3rd joint. Your palm should not change any angle, remaning calm and relaxed. Repeat a few times.
Then put your thumb (all feelings are exactly the same) on C. Repeat the process, but with 4th on B.
What do you feel?
Gyorgy Sandor from On Piano Playing p63 "avoid placing the thumb under the palm of the hand at all costs.
I did not learn playing piano from books. Although I am sure G. Sandor's wrote very fine and thoughtfull book, I am afraid I cannot comment without knowing the context, as I did not read it.
I am for the one, who believes the truth is always in between. As I wrote before, this technique has a certain sound and may very well suit certain kind or type of music, for example, Bartok, Prokofiev, etc. But to put it as a universal and THE ONLY WAY, is simply wrong. Maybe that's one of the reasons why Mr. Sandor was not famous for his let's say Schubert or Chopin renditions.
Unfortunately placing the thumb under the palm is the most widespread method of teaching scales; we must protest against it vigorously....Although it is the most agile and can move in any direction while it is alongside the hand, the thumb is totally handicapped and cramped when it is pulled in...If we force it into that unnatural and tight position (under the palm), we practically incapacitate it, and we lose all hope of achieving even, fluent playing".
What you say is instead of facing the problem, working on it, developing as many tools as it is possible for being ready to tackle ANY difficutlies you might meet on a life-long jorney called piano playing and music making, we like oistridges should put our heads into the sand (at least what they say), and pretend the problem does not exist?
I'll tell you, many greatests teachers have emphasised that developing thumb under technique is one of the most fundamental principals of piano playing.
I will tell you more. Piano playing is not only about thumb technique, but involves many other aspects, physical and mental. It is a biggest fallacy to think that ones you master your thumb (whether it is under, over, or even sideways) you will achieve fluent and even playing. That is only one of the steps!
In order to learn piano and music you need a teacher, good teacher, the teacher who will see yours (and only yours) individual problems and will methodically address them
to develop right kind of technique, which suites you, and only you. And if the teacher sees the thumb over is the only way suits you, then be it.
And BTW, speaking of teachers...
As for absolute trust in your teacher. Why? Sure, you must trust him enough to try out what he suggests, but if he tells you, drink this weird looking beverage, and I promise you that after half an hour you will be playing like Liszt himself, will you do it? Religious people make faith (blindly believing what is clearly non-sense, and even if it wasn´t how would they know?) to be a great virtue, but them much money can be made out of the faithful...
Bernhard,
The "trust him enough to try out what he suggests" to me sounds a little bit simplified.
Don't forget, when we come to a teacher we are overwhelmed with the feeling "what we WANT". A good teacher is the one who has a vision of "what we NEED".
When we come to a teacher we are like in the forest, where we don't see a next tree behind that bush. A good teacher is the one who sees all the field, with every tree, every bush, every flower, and every stone on the road. And s/he knows how to lead you through that forest and what is the best and most efficient way to get above it and see the whole field. If you do not trust your teacher sees all that--time for a new teacher.
To me it is clear that a student SHOULD and MUST do what the teacher says. However, the student should and must THINK and ANALIZE what the teacher says, and WHY the teacher says that or another certain thing. It is clear to me that student also should and must think and try ALTERNATIVE WAYS of achieving the same result. The GOOD teacher will always encourage it!
And it is absolutely clear to me, if your teacher tells you, "drink that weird looking beverage", then it is time for another, better teacher.
It´s a shame that Bernhard shirks this discussion. 
I am sure Bernhard has life outside this board as well, and might be busy. Once he has time, or feels it is nessessary, I am sure he will come to put some light.

Best,
M.