Great story idea but don't know if I should continue writing so I need help

[Prince of Spires]

As for your OP gender roles. Polynesian society was frequently matriarchal - not least because the male warriors didn't live long enough to take up administrative roles.

You can actually find something similar in old fishing communities here in the Netherlands. Not so much the entire society being matriarchal. But in a lot of families the family heirlooms would be passed down the maternal line, since the men were the ones who went out to sea and stood a big chance of not living all that long.

 

[Cady]

The Alethi society in the Stormlight Archive (Brandon Sanderson) is very close to this. In this society, men are the leaders and warriors, while women are the scholars and artists, and it is actually shameful for men to be able to read.

Just what I was going to suggest

 

[Cady]

This is a great direction to take your world building in. What you've described, however, does not sound like a story. It sounds like a context for a story. To be a story, there must be a character with something to gain or lose, and a chronology of events showing that gain or loss. I realize you might have more in mind than what you've presented here, but I can only respond to what you've written. I would encourage you to consider what people in this society could gain or lose based on how that society is constructed.

As for equality, it is not something that can every truly be achieved in a society. No two people are exactly alike, so no two people will ever truly be equal. Our individual blend of strengths and weaknesses are designed to force us to need each other to survive. Consider this: if everyone in the world were exactly like you, what would not be done? What could not be achieved? Speaking for myself, anyone needing a medical surgery would be out of luck. I have neither the knowledge nor the training to render such services to others. I know others, both men and women, who do.
Where equality is a little more possible is in the codified laws of the land. Having each person being equal under the law makes for a more equal society than one with laws that deny personhood to one group or another because of a physical characteristic. Fairness, however, is a matter of a person to person experience where each offers cordiality and respect to the other simply because they exist. Keep in mind, that those with power become terrified of losing it, and/seek to gain more for this reason.

There are many ways to think through this social construct for the purposes of story, but I recommend looking for the conflict. What seems equal and fair, but can still create conflict, since it is something your story will need. Also, why would the society chose to keep the men and women separate? Perhaps it is a matter of population control and only certain individuals are allowed to/chosen for/forced to produce children? Perhaps mating is only allowed based on certain genetic characteristics? Or maybe a woman being pregnant without permission is a crime punishable by death? There are lots of interesting things you can do with such social contexts.

P.S.
I see more details have been added since I began writing my response. Again, I like the direction your world buidling is going. I see more potential for story emerging from the context you've described.

As an experiment, you might try writing single page journal entries from the point of view of different people within this society. What is a day in the life of a military man like? Or a councilwoman? Or a child being educated? The society they have lived in has shaped them, and these little vignettes can offer a reflection of that society. These might go on to be characters for your stories, or maybe just NPCs, but thinking of what it is like to experience the world from their place within that society.

I wouldn't worry about Brandon Sanderson or any other author suing you for what you're suggesting here. He's a really nice guy and would likely encourage you to keep pursuing your writing. He did spend ten years buidling Roshar, though. Richly detailed worlds do not happen overnight.

 

[Cady]

Here is a link to a lecture Brandon Sanderson taught on worldbuilding: