You might find https://en.wikipedia.org/wiki/Women_in_Shinto to be interesting reading.
One other example is that in Japan, the arguably most formative work of literature is 'Tale of Genji' (~1000 AD!), written by a woman. This work has a Dante's Inferno like status in Japan, in that it was both formative to both Japanese language and culture. The closest you could compare it to in English is if you combine the impacts of both the King James' bible and Shakespeare's work.
This is just to show what I mean with cultural status of women in Japan - it basically started since 1000 years ago when the women's way of writing (Hiragana / vocabulary / grammar) has started to become more and more the standard thanks to Lady Murasaki, and funnily enough this is still an ongoing process with Kanji's still being dropped for Hiragana. I think you'd have a hard time finding a historical Western woman with a comparable cultural impact - Queen Elizabeth I is the only one I can think of.
Now, it is true that if you are an ambitious woman in Japan, it is going to be difficult to make a career for yourself and you will certainly face discrimination at some workplaces. But that does not mean every woman is like or feels like that.
One of the issues of inequality is that housework and child-rearing is looked down upon, not that people choose to do housework and child-rearing. There are only a few countries that have made progress on that end - see e.g. Scandinavian countries where paternal leave and stay-at-home dads are on the rise.
You don't always have a choice - because there are no or few daycare facilities available anyway, and you may not live close to your parents either.
As you mentioned, don't generalize to everywhere in Japan. I can tell you even in Kansai it's far from ideal and the daycare facilities are completely full and you need to reserve more than a year in advance to even have a chance to get your kid in.
This was last year. The situation is very much still problematic, even in cities.