They seem to forget this rule and still play the piece too fast with lots of mistakes which is very frustrating.
The reason that they play such pieces too fast and with many mistakes is not because they don't understand what is the best way to learn a polyphonic piece. The reason is that they get sucked into this state of panic when you know you don't have much time to prepare for something important, so you try to do as many things as you can as fast as you can.
They should do quite the opposite (which is counter-intuitive) - to relax and slowdown at half speed, play the separate hand parts at least 3 times in a row all in slow tempo and most of all deep down BELIEVE that in the end they will be OK.
It's not fear that keeps them playing the wrong way. It's the fear of facing the fear, the fear of having to admit to themselves that they are on the wrong path that makes them play this way.
If you struggle with any sort of deadline, please know that you are not alone. Every living organism suffers from fear.
Some flee in panic, some freeze, while some (and I hope you are one of them) stay and fight.
Here's to your strong mind!
[HT to John for inspiration]
Processional March (p. 2) by Edouard Batiste (1820-1876) from his 50 Pieces for Organ or Harmonium, Op. 24, a French organist and composer. Pedal part is optional.