I looked at the first chapter and it says that equality between the sexes are gone because only girls can get peach powers. Men are now second class citizens the world over and women took over leadership roles.
Yes but that would happen if ...
Yuuki says that gates opened all over Japan, not all over the world.
If it was only Japan, since we seen their abilities, there is no way in hell they wouldnt use their superwomen to take control of the world, so a equilibrium is needed meaning it had to happen worldwide as otherwise you would just had glorious Nippon having control yet the story acts as if Japan remains the same and the shift was worldwide.
Could be just the author not really considering things on a wider scale, the 8th Unit is being lead by what I assume is a Ukrainian (her surname is Ukrainian) and the other 2 members seem foreign as well, she also says she was "scouted by the Mato Defense Corps and would start working in Japan", the author does leave a lot out and the Mato Defense Corps seems be entirely a Japanese organization but that simply doesnt work because as I said, this would tip the balance of power towards Japan to the point they would be
the superpower.
Shikoku says that you'll have access to the resources Mato. So maybe this is what Ren meant?
She says use Mato resources freely and risk free and later says a stable supply of Peaches with Ren thinking "to turn Mato into our nation strength" and that just further points Japan doesnt have exclusive access to Mato since that is what Ren and her "leaders" desire, its obvious what would happen if a country had exclusive access to Peaches.
This does conflict with the idea access to Mato been only in Japan as if that was true, they already had near exclsuive access to Peaches even if very risky that begs the question how they are gathering Peaches.
Now, maybe there is some additional info buried somewhere in the other chapters that says why peaches are shared with other countries, but I don't want to go look.
You wont find it, there are some rather silly things since for example in chapter 2 Nei says Mato is the size of Tokyo that simply doesnt make sense, Tokyo is 0.6% of Japan and we should, pardon my ancient meme, Frodo nukes into Mato, the whole Peach thing is pretty much not talked much, we know some people got there and eaten it but we also know Pilipenko eaten one that was smuggled out by her family.
This is a manga series and being set in Japan means you get the usual centrisms, world building is a problem as characters dont say much about things and we dont see the whole picture, we dont really know much of the world outside it but we can make reasonable dectuctions, if Peache Users were almost entirely Japanese this would mean Japan would dominate the world, also Peaches have to be relative common otherwise there wouldnt be such a shift (example, in Infinite Stratos this happened because only women could pilot then and there was no actual limit to making the mechas) and chapter 1 shows a couple of girls going to school already having Peach blessings, if it was say 100 people per year there would be no diference expecially if its Japan alone but of course the manga says nothing directly about it thats a easy way for the author to not trap himself.
I will freely admit this is my interpertation as the manga doesnt directly address it.