apparently women warriors in Asian culture was pretty much normal. It's been discovered that the Naginta was primarily a female warrior's weapon.

I think you're right - in a normal situation it would have been difficult at best for women to be accepted solely as warriors. Even the Smithsonian video points out that at first the male samurai wanted nothing to do with Nakano Takeko and her group of fighting women. But the impression I got from what I've read is that acting as a warrior was on an as-needed basis. The Ten Shin Ichi Ryu site has this to say about Samurai women

The primary role of women of the samurai class had been to support the family and their husbands. During the Kamakura Period (1185-1333), women were responsible for raising their children with the proper samurai upbringing. The women of this period were allowed rights to inheritance and to bequeath property. They controlled the household finances, and managed the staff.

That sounds like a fairly typical female role and a state of things that would be mostly familiar for even a medieval European noblewoman. However, unlike their European counterparts

Women were also expected to defend their homes in times of war.

Of course, I could be completely wrong. Until today I never even knew that samurai fighting women were a thing.