So if you just apply automatic spell-correction, you have to proof read it afterwords, which is more of a hassle than accenting it manually -because it's a more cognitive work.

Correct accents in greek matter a lot. Reading incorrectly accented words make comprehending quite a bit painful and slow, my guess is because the brain expects something, receives something else and then tries automatically to make sense of what is trained to know it's wrong, slowing the whole process down.