John, in fact I was so sure because this is what the British Council makes us teach to our students. I'm a Spanish teacher of English and I trust the grammar in books, which is probably wrong or incomplete, sometimes, as your examples show.
I take note, though, of the possibility of leaving it out. Thanks.

Well, it's not redundant, it is a relative clause, and you need to use WHY after REASON to fill grammatically the subordinate clause. You cannot omit it. WHY is acting here as a pronoun, replacing the REASON, and that's why it may sound redundant, but it is not more redundant than other relative clauses with other pronouns, as: 'This is the man who saw the whole thing happen'.
'The man who' is as redundant as 'the reason why' or 'the place where', don't you think so?