I think a lot of females dislike their periods and find them cumbersome. I personally detest mine. But to be prevented from talking about them, because such conversations are considered 'inappropriate', is offensive to many, because this attitude is a hangover from a male-dominated, sexist society, where female genitalia and bodily functions were automatically seen as far more inappropriate and embarrassing than male ones, because females were supposed to be sweet and pretty and pure, and no one wanted to know about any of the actual messy bits!
To me it's similar to people saying 'I don't mind if people are gay, but I don't see why they have to talk about it - they should keep it to themselves, because the rest of us don't want to know about it'. I'm not sure if there is an equivalent attitude to trans people, but I'm sure there must be. It's a hangover from the prejudices we are trying so hard to get away from.
A similar issue (since you mention mental illness) is mental illness being seen as a big taboo, and people not wanting people to talk about it publicly. Or hiding away people with disabilities because they are seen as ugly and gross. Or cancer - there's a topic people don't like to be talked about publicly, which can be very difficult for those people who actually have cancer and want to be able to talk openly about it. I believe these taboos can be really harmful.
Of course, mentally ill people talking about being mentally ill can be triggering for those who have suffered with mental illness, or who have loved one with mental illness, and that is of course difficult, but I don't believe that the solution is to stop talking about mental illness. Maybe it's more to let the person choose whether or not to witness the discussion. So, for instance, this thread being entitled 'Period talk' could be seen as fair warning as to its contents, so people can avoid it if they think they'll find it upsetting.