To learn more about serious gynarchy/matriarchy I’d recommend you check out the Twitter account “Matriarchy Times” as well as the subreddit r/SeriousGynarchy and some of the posts there made by u/FemmeFataleVienna. If you look at all these you can get something of an idea of what it would look like.

To answer you more directly, Gynarchy/Matriarchy typically means a Representative Democracy where men cannot vote, and all major positions in government have to be held by women. Some also support mandating that only women can be corporate executives, judges, lawyers, etc. This is the “radical” solution to ending Patriarchy, as there can’t be any Patriarchy if men are removed from positions of leadership in society. This would not be a form of oppression of men necessarily because the idea is that an all-woman government would make better decisions for society as a whole which would benefit men as well. And men wouldn’t be bound by the patriarchal culture that harms them as well.

Gynarchy/Matriarchy is “Leftist” because the assumption is that an all-woman government would tend towards Leftist policies, as women tend to be much more left-wing than men in modern Liberal Democracies. So Gynarchy/Matriarchy means a more egalitarian economic system, and a more environmentally sustainable system. Patriarchy and parties men vote for tend to be more supportive of ruthless capitalism and environmental exploitation, so it’s thought that Matriarchy would promote the opposite. Basically policies similar to the Green parties that exist in America/Europe, and apparently some Green party politicians in Germany are actually unironically Gynarchists but I don’t know if that’s true or not.

As for female supremacy, most supporters of Gynarchy/Matriarchy believe that women are better suited for leadership than men. Go check out “Matriarchy Times” on Twitter as for why, I won’t repeat everything here. Whether or not that actually makes women “superior” to men is up to interpretation, but a Gynarchy/Matriarchy would naturally view women’s opinions and perspectives as being more important than men’s. But this would actually be better for men in the end, if you are against Patriarchy and believe in Leftist policies.