It seems that every piano teacher has a different view of just how much worth these exercises have.
Is there merit to having beginners start with these exercises and work through a sizeable chunk of them or is it more of a case of, begin with your training and when you start to struggle with a certain part of the technique then do the exercise that develops that part of your technique.
The best place for the Hanon exercises is on a bonfire, IMHO. They have zero musical value, and negligible technical value either.
Czerny's many collections of exercises have a lot more value, so long as you realize that they are based on 200-year-old keyboard technique and written for 200-year-old pianos, which were significantly more "fragile" than modern instruments and usually had a lighter key action.
By all means use Czerny in conjunction with more modern studies - but it would very limiting to use only Czerny. A diet of Czerny and nothing else won't teach you to play most of the piano music (of any genre, not just classical!) written in the past 150 years or so.
There is a basic dilemma in music teaching for any instrument, not just piano: do you teach how to play the instrument or how to play particular pieces? Expressed like that, it seems rather obvious that the first option is better, but all students need the gratification of "playing pieces" as well as learning the necessary technical skills. The ideal balance between the two is very student-dependent, of course.
Technique doesn't come from a finger exercise. It comes from knowledge of physics, anatomy and mechanics. Your arm is a "machine" of levers, pulleys, rubber bands, fulcrum and there are proper and improper ways to move. Hanon nor Czerny can give you that. No matter how many hours and decades you spend playing them, if you are moving incorrectly, you are only hard wiring improper movement into your brain.
Learn to move properly and a facile technique will just fall out of your hands with no exercise, warmup or need to practice. Do you need to practice riding a bike? No, once your brain hard wires proper movement and balance, it is there forever. The arms are no different. Once a bad habit is hard wired, like the first time we touch a piano, it can take a lifetime to eradicate it.
If you learn to use the incorrect muscles as most pianists do, or non-existent ones as Hanon and Czerny espouse, you will hard wire improper movement into your brain and forever struggle to beat your hands into submission. That is why teachers spew "endurance and strength" because they don't know the laws of physics which result in effortlessness.
If you've ever run in a three legged race, at first it is a struggle but once you and your partner sync up, it is easy. However, if one of you doesn't have the mental acuity to sync, there will be an imbalance and there is not much YOU can do about it.
Playing from the wrong muscles creates a myriad of imbalances and teachers think you need "strength and endurance and more practice" to fix it. No, you need a better teacher.
Playing the piano is all in the arms. The fingers are the conduit between the brain and piano. Hanon can be wonderful if you have an ergonomic technique but, if you have an ergonomic technique, you don't need Hanon.
If I see Hanon at a garage sale for a nickel or quarter, I buy it so no one else does. Then I use it to start the fire in my smoker. Mmmmm, 16th note burgers. Gurgle, drool . . . . (my Homer Simpson impersonation).
If your teacher doesn't know how the abductors affect the flexors or, the importance of pronator and supinator muscles, or the dangers or ulnar and radial deviation, then you've probably already developed bad habits.
Likewise, if a teacher teaches to cross the thumb under the palm . . . RUN!!!!!
I'd go with the second option, because I think exercises can be very boring and discouraging for beginners.
I have always learned by playing pieces. Each one of them needs different techniques, and by studying those parts you develop them as well. So, why do an exercise instead of studying a part of a piece that employs the same technique? Besides, it's much more fun and gratifying, and I believe the result is not so different.
Of course, some people like to do exercises to improve their technique faster and play more difficult pieces sooner, this is personal.
I believe in general it's better to learn through pieces, not exercises.
