Our district has those questions on the yearly form. (date of first period and date of most recent). We always make up the answer for most recent - just because we never remember. I would have no hesitation telling my child to lie about it if there was a concern.
It never crossed my mind to be concerned about it, but I can see both sides of the debate. It does get turned in to the school, but only to the Athletic Trainer. The coaches/teachers only see the shortened emergency form. I don’t know the Athletic Trainer’s background or qualifications, but she is medical-adjacent enough for me to not have a huge issue with her having access. But - I don’t know who else has access, and I do understand why people don’t like that.