You could, doing so would likely increase utilization of the tampon policy meaning it would have a larger benefit for the same cost.

I suspect if you tried to in schools you'd end up with a weird Title IX ruling that claimed that not having a girls room (regardless of availability of a unisex restroom) is sex discrimination in a way that not having a boys room isn't, because equity. They have a bad habit of taking that kind of approach, see school sports.