As a woman whose period symptoms are not currently severe, but were for several years...
I used to have one morning per month when I could barely move from the pain and the naproxen I took to make it bearable would prevent me from concentrating on anything more complicated than children's cartoons. But I stumbled into work and faked productivity anyway because I only got 10 days of PTO - vacation and sick days combined - every year. If I'd taken time off to deal with menstrual pain, I would never have had a vacation at all.
And I'm a lucky one. My menstrual symptoms were only moderately severe, and they usually eased by mid-afternoon. Given flexibility, I would probably have worked remotely, taken a couple hours off in the morning (only answering urgent communications), and worked a couple hours later in the evening to make up for it (instead of bolting out of the office at 5pm sharp because being there sucks even harder when you're bleeding like a stuck pig and your intestines are revolting against your OTC meds and you generally feel like you've been hit by a bus).
Probably 25-30% of women (even without additional medical conditions like PCOS) have more severe symptoms than I did. I do think it's valuable to take menstrual problems into account and I would appreciate a workplace that is sympathetic to women who suffer severe menstrual symptoms.
But I hate that this particular proposal creates gender differences in the workplace. Yes, it feeds into the wage gap and into reluctance to hire women. It also creates weird medical privacy issues as several other posters have pointed out.
The right solution probably involves some combination of schedule flexibility; remote work; more sick days for everyone, not just women who happen to menstruate; and separating sick days from vacation days.