Play good and bad and ask what was wrong? How would she know? I'm speaking technically what is making the difference, and i'm assuming that the difference between good and bad playing is the performers technically facility, depite how good an ear they have.
To me I have believed that producing good or bad sound almost totally depends on what you actually hear from within. We do not memorise what sound to produce in a given piece by memorising PHYSICAL action. These things are observed, studied, practiced and refined at first but we eventually put them aside and focus on the sound production. Or what action moves towards a better sound production.
We cannot let students get stuck in that observation of the physical action however. We must push them to listen to themselves and forget the physical action for the moment. If they know what an even sounding scale sounds like then they will see that if their fingers fly all over the place that ideal even sound is disrupted. In a string of single notes there will be always one note frustrating the student or causing discomfort, the teacher must pick it out and demonstrate to the student tools to deal with these difficulties. This is sometimes too interlectual for younger students to remember and apply so then you just spoonfeed them instead of trying to explain it.
A problem can arise if observing bad sound production is not worrying the student enough to make changes to their physical action. I have some younger students for instance who just like to play the notes, like typing keys on a keyboard or playing computer games, this button then that then these two. Sometimes young children just do not care enough about the right and wrong sound, it doesn't interest them. This is where musical immaturity can effect their progress. All good musicians care a great deal about the sound they are creating and constantly listen to themselves closely when they play, if this habit is not started somewhere then you are not going to get very far in the quality of your playing. You also cannot force this care that isn't the teachers responsibility. I am happy that none of my students now are like that though *phew*

The physical action is controlled by what you hear, that is hard to describe but the beginniner student should accept it as true. Eventually they will realise what it means. For all experienced pianists if we say to them, play the opening chord of Beethoven's Pathetique Sonata, that Cminor chord, they all can imagine the shape of their hands and how they would excecute it. They all can sense the force that their body will push into the keyboard before doing it. But a beginner has no concept of this. They know it would sound loud but they don't have that automatic response or sense of their physical action as a more experienced player would. This is why we must let them observe action and sound as much as possible and practicing making decisions.
The teacher must demonstrate to the student so that the student can start to making decisions and observe "right" and wrong" sound and action at the keyboard. I can imagine a student who never listening to themselves while playing will play like a robot, relying on physical action to produce the sound rather than their ear which controls their sound thus controlling the physical action.
In such a subjective issue as piano playing we have to be guided by compasses, good and bad, rather than laws which are set and inflexible (eg: the old fashioned idea of keeping constantly curled fingers while playing as opposed to flattened fingers which changes shape but always maintains a "relaxed" form.) This idea of relaxed is very subjective and fits perfectly with music, it is a flexible tool to use yeilding with the bends and contortions of music. So too is the tool of LISTENING to yourself a subjective tool which moves alongisde music all the time. We can only improve our listening tool by being exposed to observing good/bad piano playing both in sound/action. Sound is usually much more obvious, action is harder for beginners to pick out of course and thus the teacher is responsible for laying some foundation for their understanding as to what is good and bad physical action.
What i am asking, is whether 'correcting' flying fingers at an early stage makes any difference or not?
If correcting it changes the sound then it will. I have a beginner student who first kept her fingers that didn't play curled up into her hand. It did not effect the quality of sound because the pieces she played at first did not demand that they stay open. Of course when we moved to harder pieces which demanded more of the fingers to work she hear how the sound was disrupted with her curled up fingers and how uncomfrotable it was, so we worked out a way to make it feel more comfortable with a few experiments and at the end of the lesson she knew logically how to deal with the new idea of using more of her fingers and knew how to work against her fingers curling up even though they did now and then but she knew, oops! shouldn't do that because it makes this bit harder.
Correct things if it effects the sound of pieces she is studying. The teacher must make a judgement as to if it does or not.
I very seriously question this assumption, because i do have students for which it is not a problem, 5 years old. Just how much 'strength' does it take to press down a key? IMO, the issue has absolutely nothing to do with strength (unless you're playing Rach 3rd), but the ability to co-ordinate the fingers to act inter-dependantly.
Notice that I said sometimes in my assumption not always. Finger independance, using the finger to press down a note with only the isolated action of the finger is not something I would teach beginners. But being able to control a string of notes and play it even and balance utilising the 5th is very important. How you do it of course there are many ways but I would avoid asking them to find energy isolated in the muscle of the little finger alone to play the notes. That is why the idea of using the weight of the hand to give the 5th more strength is a good tool to give beginner students. It is only a tool which i have found beneficial for students with weakened fingers, know how to use the weight of the hand to support weak fingers. Of course these fingers develop in strength but all beginner are faced with the weakness in some fingers at first. I don't like to shrug my shoulders and say, well as you learn more pieces your fingers will get stronger. I like to think we can take action and actively impove their fingers. Hanon then to me have been like Weight training for the fingers of my beginners rather than scales. Alterations to hanon excersises to practice and strengthen students personal physical weakness is a matter for the teacher to consider.