Here are the main reasons why legato fingering is advised:
- There are many times when a note (or notes) is sustained through a pedal change. Sometimes this happens during a harmonic change in the underlying notes, while the melodic note must sustain through to the new harmony. The sustained note can be anywhere, but usually in the melodic and bass lines.
- Similar to #1, during many pedal changes, you will want a hazy mixture of the old harmony mixed in with the new harmony just before you make the clean pedal break. In slow, melodic works like this Beethoven, this is the preferred pedal technique. In slow motion, it looks like this: hold down the new note(s), slowly lift the pedal just enough to clear out the old notes, then bring down the pedal to hold the new notes. In fact, all pedal changes are like this, but at varying speeds depending on the passage. Beginners sometimes mistake this to mean that the pedal comes up before the new harmony, which results in a break in the sound.
- At the advanced level, playing the piano is more about transferring weight around, especially in melodic passages, from one finger(s) to another. This is best achieved by shifting/moving that weight from one finger to another, rather than lifting off after every key stroke.
- Some works, usually Baroque and early Classical, require that you not use the pedal, or very little. For example, in the Bach fugues, you must hold down long notes with your fingers.
- Many will argue that there is a difference in sound between a key held down by the finger vs. the pedal. As the hammer strikes the string, and the damper goes up at the apex of the hammer strike, the reciprocal downward return motion of the damper has an effect on the string, causing the string to vibrate less freely and inhibiting the sound. Some will go further by saying that the damper felt actually touches the string ever so slightly, but enough to shorten the sustain. Anything that shortens the piano's sustain is a bad thing. You can test this theory on a good, grand-sized piano with the lid up (so you can hear clearly). Strike a ff chord with both hands, once with a staccato motion, and once by holding the keys down. Pianists often use the former technique to achieve a pedal-ed sforzando effect where the burst of sound is more quickly muffled than the latter method.
Regarding small hands, if you can reach an octave, you're fine. There are/were many with very small hands (de Larrocha, Hoffmann, every eight-year-old Curtis and Juilliard prodigy in history) that play(ed) very well indeed. The trick is to relax. Those with small hands have to maximize every bit of agility and flexibility from their hands, and the only way to achieve that is through relaxation and solid technique.
Regarding the comment that no one in the audience will hear the difference between x and y, that's mostly true, but what they will notice, and react to, is the overall effect of the hundred things that you're doing. No one cares about, or hears, all of the excruciating details that you and your teacher have prepared, but all of that work does mean something when you transfer that work into performance. Listening to two performances of a very simple work played by a professional and by a very good beginner, a casual listener will not know many of the technical difference between the two, but will feel the musical effect of the pro's performance. All of the agonizing details that you've worked out in practice translates to a kind of artistic expression in performance that has much less to do with crescendi and diminuendi and curved fingers and pedal technique than you realize.
On a final note, beware of those that advise that you do things the way you want to, especially in spite of established techniques. At the beginning of your studies, many things will seem counter-intuitive, and your own common sense solutions will make much more sense. You'll even see contradictory examples of what your teacher is telling you. Everyone struggles with this throughout their careers, but succumbing to this intuition is especially dangerous at the beginner level. There are many things at the advanced level that people can and do disagree on, but at this level, it's best to follow established teachings. If you really think about it, pressing 88 flat levers with your fingers is not a natural thing to do. Hundreds of years of thinking and practice have gone into how to do this unnatural thing as beautifully as we do now. Take advantage of it.