The true purpose of a formal piano training is to explore the maximum musical potentials of the pianist. Through a series of finger exercises (scales and arpeggios) to experience in musical expression in the form of dynamics and so on, a capable pianist will be able to bring out exactly how they hear the music out into the world. Piano playing that is technically brilliant but lacking musicality and originality of thought isn't better than getting your computer or MIDI keyboard to play a musical piece. In fact you will be stunned at how many artists have so different interpretations of the same piece that they often sound like totally different pieces! Their technical efficiency is only a tool they use to get the music out of their mind and into the world.
Therefore I conclude that it is most important to train the mind before the fingers. The painter must have a clear vision of how the picture will look like, then use the specific brush strokes that will bring this picture into real life. Same goes with music, a pianist should have a clear thought of what they want to hear, before attempting to tickle the ivories.