I have delved much into the concept of playing from memory in recent years. Just think that until recently (we are talking about perhaps not even a couple of years ago) I couldn't remember anything by heart, because even as a child I was never "educated" in memorizing pieces. Some famous pianists, the most representative of whom was Richter, used to perform pieces with the score in front of them even in concert. There is a famous phrase of his in which he justified his approach with the score because, by avoiding memorization, one could allow oneself to use that time to learn more pieces to add to the repertoire. Naturally, there are pros and cons in both versions. Currently, despite having managed to memorize a good amount of pieces in my repertoire, I would not perform in concert without a score. It has often happened to me in concert not to even look at the scores, but now my issue is psychological in nature, as if I needed the score. A "phobia" that I should have fought when I was little.
In any case, for the memorization of pieces, there are several types of approaches that are complementary and exploit different types of brain functions. There is mechanical memory, photographic memory, auditory memory, and harmonic memory.
The mechanical memory is what most obviously helps us with passages involving leaps. The arm moves a distance that is remembered by the brain.
The photographic memory is what helps us visually remember the movement to be made on the keyboard or the score.
The auditory memory is what reminds us how each measure of what we are playing sounds, dividing the sound into two hands (typically for those who enjoy possessing perfect pitch, this memory overrides all others, but the gift of perfect pitch is something that statistically few possess).
Harmonic memory is acquired by studying the piece harmonically as we practice and remembering all the tonalities it passes through. Unlike auditory memory, which functions mostly on the horizontal line (that is, the melodic one), harmonic memory plays a fundamental role at the vertical level, namely on the chords.
By combining all these memories, in the end, one manages to start from the beginning and reach the end of the piece without hesitation, as each supports the other automatically.
Naturally, developing and training all these types of memory takes time. If before, by reading the piece, I could get it up and running in 10 days, studying a Chopin ballad from memory requires much more time.
The method of closing one's eyes honestly seems exaggerated to me as well, so I agree with Paolo, and perhaps it is unproductive because it forces you not to use an important memory, which is the visual one. In my opinion, these things are more like special effects in movies like "The Legend of 1900" and similar.
Speaking of memory, one must reach a point of awareness where all these memories intervene automatically without having to think about them, so that your mind can concentrate on the most important thing: being in the music and interpreting the piece correctly. Naturally, memorization is the last of the processes in preparing a piece, at the base of which lies good technical preparation. Not having good technique puts you in a position where you fear a particular measure, and that is the moment when concentration is lost and there is a greater risk of forgetting or making mistakes. Technique is the engine that allows you to travel down a downhill road just as easily as a very steep climb. If the engine splutters, it is very likely that you will manage to travel on a downhill or flat road, but the idea of taking that car on a road where you know you have to climb a long and steep hill might distress you.
A pianist's preparation impacts many variables, as you can see. What the audience hears is the final step of a long work that engages the pianist's brain 360°, and many cannot even imagine the work—most of the time stressful and painstaking—that lies behind the preparation of a concert piece.
