While healthy ergonomics should of course be a concern for ANY physical activity (piano, ballet, soccer), etc; it's my opinion that there should be some consideration whether or not this it should be considered the chief concern.
First of all, I think there is an erroneous conflation in the piano teaching world of the terms "Hand Position" and "Technique". Many people generally assume they are the same thing, when in reality this could not be further from the truth. "Position" implies something that is fixed - yet a healthy and fluid piano technique is anything BUT fixed: When a student reaches an octave their hand is more open, and fingers flatter than when playing a fifth; When the thumb tucks under the third finger during a scale, the hand contracts slightly and the wrist and forearm raise a bit, when you want to get a singing tone in a Chopin Nocturne you play on the cushion of your fingertip, making your finger flat. etc. A technique that is fluid and truly does justice to the music is always one in which individual motions are giving way to new ones every second with effortless evervescne.
This notion aside though, I think it is possible to define - if not a fixed position - at least some guiding physical principals about what our hands should be doing at the keyboard. (These guidelines are generally for beginners only, once one gets into even intermediate repertoire the music demands too many counterexamples to let them be rules) :
1.) The hand should be supported by the knuckles that create the bridge; they should not be collapsing. The individual finger joints should generally be out (although some collapsing is not the end of the world). From all of this joint support, the general shape of an arch is formed - which is the strongest shape in nature.
2.) The base position should be natural. What is a "Natural" position?: Find out by standing up, letting both arms drop with dead weight to your sides, and observing what your hand looks like when it is in this completely neutral position: The fingers are only moderately curved (Unlike the death-grip claw that we are often told is the basic feature of a good technique).
3.) There should be relaxation. Obviously, not total relaxation, because tension is what creates muscular contraction, which is what we need to play the piano. But there should not be extra tension. (Extra tension is basically anything we are doing that is not NECESSARY to make the sound we want: Tightening and lifting of the shoulders, making squinty faces, and just a general sense of pressing and holding more than necessary for the task at hand.)
4.) There should be alignment. When playing the middle fingers (2,3, and 4), one should feel they are not separated from the the arm, but that the wrist and whole arm are cozily and supportively behind them like loving parents in a "line". Avoidance of anything "twisty". Leon Fleisher says that pianists spend too much time thinking about "fingering" passages, when we should spend more time "arming" passages, as the fingers are just the ends of the arm broken up.
All that being said (And this is where I stand ready for the fire that awaits me in the comments section) if these things don't happen, it's okay and not the end of the world - particularly for young students, and ignoring it might even be beneficial. I do believe correct technique should be explained, and regularly and gently encouraged, but should not be dogmatically insisted upon, at least on the expense of other things, for these reasons:
1.) The core feature of a healthy technique is relaxation and freedom. But by continuous berating and insistence upon maintaining a certain position or doing specific things with one's hands, it is possible that this stress and anxiety may cause a student (Who is still developing phsyilcaly and psychogocially) to paradoxically become even more tense, all in the name of relaxation and "proper" technique. (I studied for one semester with a teacher renowned for "technique". In her impatience, frustration, and exceeding desire to get her students to play freely and without tension, she would scream at us to "RELAX!!! RELAX!!!, with a fury and vexation so great that scalping her cat could not have evoked more rage. Needless to say, that particular semester, none of our "techniques" got any better.
2.) For some students, particularly very young students 7 and under (and especially those who have ADHD), it does seem quite impossible for them to maintain a "proper" hand position outside of the very moments you are asking them to do so. In order for you to work on this, you would need to spend every second of every lesson making sure that they have proper form - and the only hope for home practice would be to make use of some contraction like a hand rail placed alongside the bottom of the keyboard (Which was popular in the Leipzig schools the 19th century). I think that aside mentioning it a few times, or demonstrating what it should "Feel" Like (Perhaps with one of those cute ladybugs going around in circles), you just have to let it go and teach. I know some pedagogy is still very strict about this: Some teachers will assign only five finger patterns and nothing else until a student has learned to play with "proper form". At best, I believe, that is a demonstration of patience and faith that I've yet to discover, but at worst I believe will cause many students to drop study and become uninspired early on, when it could have easily been easily avoided. It may be strict, but strictness is not necessarily indicative of good teaching.
3.) Although playing the piano involves physical processes, it beings and ends with non-physical processes - things that are in the mind and ear. This contrasts with sport, which also begins as a mental process but in which the end result is purely physical. Few people go to a piano recital to watch the way the fingers move across the keys (At least not in the same way one would watch a figure skater), they come to marvel at the sounds made by the fingers, which, although related, are two separate things.
Playing a musical instrument successfully makes use of a mysterious Boomerang effect; we pre-hear, pre-want (all in the mind), then the Boomerang flying through the air is the actual playing (the outward manifestation of what we hope to achieve), and the sound that results from that paying and reaches other listeners and ourselves, is the return of it. While the Boomerang successfully flying back to us depends on how it flew, how it flew is a direct result of what we did before we threw it. This can be likened to the going-on's in our head before we play in the moments that don't even involve our fingers, or any technique. All of the impulses, thoughts, and decisions we make before we throw will eventually determine if it comes back.
Given this, I think that there are aspects of music education that are much more primal than technique and hand position - things like rhythm, pattern-recognition, dynamic perception, emotional connection and love of music. If these things are all firmly in place, then not only does it become easier to work on technique, but there is a greater chance that one's technique will form more naturally, because when one has a clear idea of what they want to do in the inner ear, It is easier for the physical components to take shape in a natural way. If a student can recognize the hallmarks of a steady pulse, beautiful tone, and eloquently shaped phrase, they will discover that it is impossible to create these things without the proper physical components. But, if we work backwards, if the primary concern is to be aware of what we are doing physically, it is impossible for them to know and hear the why of what they are working towards, which will make it all the more difficult (and boring).
So when somebody is worried that a student has a poor hand position, it may be true that it needs addressing and drilling to fix. But this should never come at the expense of the more important (in my opinion) aspects of study. If the student has no sense of what a pulse is, no aural or reading abilities, poor rhythm, no interest or love for music, no inquisitiveness about symbols and words on the page, or an inability to focus, then a "poor hand position" is the least of their worries.