The penitent can do what they want, of course, but he is advised to not write it down, and advised that it is never a part of forgiveness to do so nor to dwell or obsess
In a piece on why can’t Catholics do confession via email, it can’t be done. The Church doesn’t want anyone writing down their sins, and for the reason depicted in this story
I don’t know where you encountered this idea, but there’s no basis for it. Any person is free to write down anything he pleases about his own sins, or life in general. The Church has no rules about this, in connection with Confession or otherwise.
Of course obsessing about the past is not healthy, but this has nothing to do specifically with the Sacrament of Penance.
The reason the Sacrament of Penance cannot be received by e-mail, or telephone, or Skype, or by letter, is that the penitent and the priest are not physically present to each other. It has nothing to do with any prohibition of writing things down.
In fact, if a person who cannot speak for some reason wishes to confess, he can do it in writing. If he speaks a foreign language, he could have a trusted friend write down his sins in a language the priest will understand. (Of course, the friend would be bound by the Seal.)