i agree with everything that Bernhard had said.
i started quite late, 14, i'm now 22 years old and my repertoire includes works like the schumann concerto, prokofiev op. 4, chopin 2nd sonata, a few chopin etudes, etc.
the brain really is a wonderful organ. the mind is an amazing gift. i firmly believe that technique is all in the mind. technique for me is the ability or the skill of being able to produce different sounds and colors on the piano, whether in fast and virtuosic music, or in slow and lyrical ones (or in any other character).
i used to practice with open fingers, really loud, because it was what my teacher told me to do so. he says it would make my fingers strong. i did it for a year and i injured my left hand, i wasn't able to play for a few months. now, it's healed but unfortunately it's already there, the injury. so i had to think of a different way of practicing and playing.
Sound-
i realised that using the fingers to make the sound (whether loud or soft, slow or fast) can be very tiring since it had very small muscles. it also did not a make a very good sound too. i started utilizing the whole arm, with a good approximation of how much weight to put in every time, whether it'd be a running passage or a big chord. it's more relaxing and natural.
Dexterity-
everything that deals with speed is mostly for my mind to achieve. i think of the changes of positions of my hands: fast, fingers positioned on the correct notes of the next position as fast as possible, relaxed arm muscles in the process of changing a position in order to avoid making accents.
i also think in groups of notes, the larger the better, although not large enough to get myself lost.
Memory and being able to play fast-
everything has to be backed up by a very good memory of the music. not just memorizing by sections, but in the smallest details possible. i'm not saying that i was able to do all the techniques in memorizing, but i'd do anything (sometimes i make or improvise my own) that i think might help with my memory like playing in the air a whole piece.
for me, memory has speed! the faster i can think of the notes (almost automatically, because we also memorize the feeling of the muscles working), the faster will my fingers respond.
not only will I be able to play fast, memory slips are avoided. if ever there would be a memory slip, i would know exactly where to go or what to play to get out of it. that's because i tried to know the music in many ways i could possibly do.

What about clarity like in a mozart?
i practice open fingers but this time NOT loud. i don't try to achieve stronger fingers but i just feel the stretch that tells me that i'm currently playing this particular note with this particular finger. so it actually has to be very slow, like 60 to a 16th note.
Starting at 14-
it's quite late. i never played classical music before that time, and piano was just a hobby. the only disadvantages i experience right now is that i learn pieces at a slow pace (actually, i like it that way), and i sightread very slowly.
i think that those who started at a very young age had the advantage of collecting certain hand positions, certain passages or groups of notes, that pop out of music pieces, and everytime it does, they'd know exactly what to do, what to think of.
it's never too late.
good luck to your students and to you, too.