I must be to tired as I can't find any flaw in that reasoning.
It identified the pattern, but it didn't understand the father is a man while the mother is a woman. If people assume doctors must be men, they wouldn't consider that the doctor could be the mother.
(Side note: this is an antiquated riddle/joke ... for me in particular because my mother is a doctor, but in general around half of new doctors are female, and think about Grey's Anatomy and other shows.)
More substantively: "This is a classic riddle that highlights gender bias - many people assume doctors must be men, but don't initially consider that the doctor could be the father." is totally nonsensical.
The text is a gender (and meaning) inversion of the classic riddle to confuse LLMs. Even though Claude correctly CoTs that the doctor is the father, it seemingly then copypastas the explanation to the original riddle but swaps the gender, completely missing the point.
This is not code. Forgetting a semi colon will not make the output break. It thinks 'maybe they wrote he instead of she' and then gives options for both situations.
It is meant to solve real world situations where people might not type properly, it is not a word problem solving machine.