While I won't say you should never move your elbows, you most certainly shouldn't make a habit of using your elbow in all circumstances as you seem to suggest. The mere mention that you cannot do anything without using your elbows calls into question your whole argument. The second you start moving your elbows is the second you introduce tension to your whole arm, simply by flexing the muscles necessary to move the elbow, because those muscles cause your shoulders not to be relaxed as well.
The point I made is not that your elbows should be unmoving and unflexible, but that they should hang loosely and be relaxed. If this happens they will breath with the music and be relaxed, allowing you to more efficiently do anything you want to do on the piano.
I think your argument displays a natural prejudice against elbows! First you make a rather tepid statement in support of their usage; then you say even to use them at all is to "introduce tension" and destroy the support of the shoulders. Nothing could be further from the truth. All the parts have to work together; if for instance using one's elbow is causing unhelpful tension in the shoulders, it is clear one is doing it wrong.
Elbows should not be passive as you describe in the second paragraph. Rather, in many passages, and many types of passages, they have to take an active role. The confusion comes, I suspect, from the notion that there is a "left" or "right" motion, in other words that they are to move in a linear way. However the true motion of elbows is best described as "clockwise" and "counter-clockwise."
A good passage to show this would perhaps be the left-hand of Rachmaninoff's Prelude in B-flat, op.23 no.2. Playing very fast, you have to make large leaps and have the whole hand be ready to play the next thrust of notes. The only way to do this, obviously, is to not cross over with just one finger, but to play in the so-called "thumb over" technique.
This is the very technique that requires, in this case, a clockwise motion of the elbow. There is nothing passive about it; the elbow is not hanging "relaxed" and just following the wrist, nor is it stationary. Relaxation is certainly a precondition, but move it must.
Another telling passage is perhaps in the second of Prokofiev's Visions fugitives, in the 10th bar; a slightly awkward right-hand passage that is greatly relieved by a counter-clockwise motion of the elbow to get from place to place.
In general, the kind of technique that Rachmaninoff & Prokofiev used greatly favors this motion, especially Prokofiev, who often asks pianists to use all five fingers in sweeping scales and arpeggios. Forcing the pianist to move his entire hand from place to place, is forcing him to use his elbow!
Walter Ramsey
I thought of another passage - actually several, but one which is perhaps more illustrative - and that's the coda of the second Ballade of Chopin. On the last page, when the right hand has the towering arpeggios combining chords and single notes, the only way to play the passage without tension is to use the clockwise/counter-clockwise motion of the elbows. Try it...
Another passage very similar would be the last E-major peroration of Ravel Toccata, also in the right hand.