If periods weren't a thing, would women carry purses?

More generally, my question is: why are purses typically seen as a gendered item? There doesn’t seem to be anything especially female about carrying things, so why is the packaging of those things so subject to society?

A few theories I was considering…
– Most women have that time of the month they want to carry around certain things discreetly, so purses just become a habit? (I know that was when I started carrying a purse.)
– It’s more acceptable for women to wear makeup (actually kind of expected)—and makeup may need touching up throughout the day?
– Most men’s pants tend to have better pockets than most women’s pants, so men can more easily get away with no bag?
– Or, since “masculine” bags tend to be work-related bags (briefcases, messenger bags, etc.)... Perhaps men always did have bags, but they were tied to occupations (before it was socially acceptable for women to join workforce)?

I don’t know why this is interesting to me, it just is. Maybe it’s interesting to you too? What is/are your theory/ies?