I'm sorry, but if you're going to tell me that for a young hand to span a third between 3 and 4 (while leaving the 2nd stuck right against the thumb) is "mechanically efficient" I have no idea what planet you come from to possess such a hand. You sincerely feel it's less natural to open the space between thumb and 2nd finger to a third than to do so between 3 and 4? This is utter nonsense, beyond the nature of "opinion". There is no more natural place to open out than between thumb and 2nd.
You are still ignoring the articulation: staccato. I would not disagree that in a legato passage, 3-4 would require a greater stretch, however, in staccato, the forearm is simply displaced. There is no doubt this is what her child is doing because at that age, the hand is incredibly small.
Mechanically, the 2 functions much better when it retracts: it has it's own muscle AND a single tendon attached to the digit allowing it to be very mobile! (Contrast the 3 and 4 which share a tendon.) However, playing the piano simply be retracting the fingers is physically inefficient because of the lack of mass of the digits. This is where the "arm weight" school of thought came in, to correct previously held beliefs that piano playing was a digit-oriented instrument.
But one aspect that you are probably not aware of is that in order to stretch/increase distance between the thumb and 2, either:
1. the thumb extends (moves away from the palm)
2. the index finger extends
3. or both the thumb and index finger extends.
When the index extends, it increases the distance from the center of mass of the entire hand. This drastically decreases the effectiveness of transferring the momentum into depressing a key; i.e. the finger is not curved. The reason pianists learn to play with curved fingers (whether taught to or they figure out on their own) is because of anatomical reasons and the efficiency of the transfer of mass. This transfer of mass is what allows a finger - that is attached to the hand that is attached to a forearm that is attached to the torso - whose mass is much less than key (a finger will never alone depress a key until the mass of the key is less than the finger) to depress a key.
You're basically suggesting that it's easier to do a vulcan salute (where 123 are bunched up and 45, with a gap in the middle). It certainly is not. It's easier to keep the four fingers together and bring the thumb away.
No, I did not suggest to do any sort of Vulcan salute. There is no stretch between the 3 and 4; it is kept in a neutral position with only a slight lateral shift of the forearm and perhaps a slight outward wrist abduction to position the finger over the key. Kaplah!
So, Liszt's professional musician father never once would have helped him? You are making up facts. This is total nonsense. Not to mention the fact that the average child is NOT a genius. Take even a million kids and give them a piano without tuition and see how many will become great pianists. To base general teaching on such a model is simply foolhardy.
You implied that Liszt is the result of having teachers. He received some instruction from his father but consider that by all accounts, including Czerny, Liszt had an unusual technique which included strange fingerings. How could Liszt have such an unusual technique if his father gave him lessons? His father's influence at the technical aspect of the piano was minimal which is why he searched for a teacher for his son. It was Czerny who set out to correct his supposedly faulty technique. A slight digression which requires some deduction but considering Czerny's idea of technique, contrast with the ability displayed by Liszt = Czerny did not create Liszt. Liszt created himself at the age of 21 long after receiving lessons from Czerny.
Regarding this bilge about how to avoid the single most important finger in piano playing, try this instead.
The exercise in the video is simply an exercise. I've done it years ago thinking this was a way to practice and improve my finger dexterity. However, it does not work very well for the intended purpose of playing the piano. The reason is pretty simple, it only works as an exercise because it focuses on moving the fingers but the finger are not the only things that must move to play the piano. On top of that, those are not motions that can be done for extended periods of time as the muscles will inevitably tire.
The alternative to those specific movements occurred when my fingers began to tire. I didn't realize it at first but to compensate for the burning sensation, my forearm started to rotate slightly as did my wrist start to glide on its joint. Immediately, with some fine tuning, the burning sensation went away and the movements just seemed natural as opposed to keeping a still arm and wrist.
Any kid can get their fifth to work when they start from this approach. It's when they try to jam the whole arm through the fifth that it's perceived as being weak. While I think Roy Holmes attributes rather too much to this approach, it's a great way to feel what the fingers are really able to do.
The approach is moot and is not in anyway feasible. A student who is just learning how to play the piano will not have the dexterity to play in such a way but mostly because it is uncomfortable to keep the wrist and forearm still while moving the fingers in such a way. The fingers which are capable of those movements are very inefficient for depressing the keys using those movements. That approach is very similar to the finger equalization school of thought where each finger has presumed weaknesses and exercises are there to equalize them. Fingers do not have weaknesses, only certain strengths (not just in the muscular way). It is up to the body of the person playing to figure out what those strengths are.