I like Ms Scott taking Mr Scott to task for stereotyping, but it does beg the question, in a genderflipped universe would the stereotypes be flipped too?

This doesn't actually seem to be a fully genderflipped universe. Kirk has found herself coming up against sexist attitudes (like with Turnabout Intruder, this strikes a strange chord for happening in the 24th century, but, whatever). Looks like the genders of the major named characters got flipped in the last two generations, but the indicators are that it's otherwise similar to the main universe.