Yamato is going away with the Strawhats at the end of the arc, regardless of whether she's in the crew or not. Absolutely no chance she's a Shogun figure. This is One Piece- when a character expresses a desire of escaping from metaphorical and in this case literal chains to go on an adventure, they get to do it. Not get stuck in a role they don't want.
Yamato never said she wanted to go on an adventure. She said she wants to sail on the straw hat's ship, not because she wants to pursue something, but because in her head, since she's Kozuki Oden, it only makes sense for her to travel with Luffy. This isn't a genuine desire, it's something that arises from backwards thinking and is completely tied to her Oden impersonation. Take away the impersonation and there goes her need to feel she has to travel with the crew. She's not like Carrot who actively seeks adventure itself, and she's not like Oden who sought the same thing. She merely wants to live out a fantasy she doesn't understand.
-Speaking as Kozuki Oden, I'm a man
-Speaking as Kozuki Oden, I am Momo's father
-Speaking as Kozuki Oden, you should be giving me a ride on your ship
She's merely read his life story and has literally imprinted everything that happened to him, onto herself. Making both her past and her present the same as Oden's. Which is stupid of course.
She's been stuck on Onigashima for 20 years. Exploring the vastly different climates found in different regions of mainland Wano should be ample adventure for someone whose ambition is as weakly founded as hers is.
Yamato-as-regent doesn't answer that in the slightest. She's not, barring some shock revelation about her mother, a Kurozumi. She has nothing to do with how they were treated. I doubt the people of Wano even know who she is.
You're totally missing the point. Yamato's not a Kurozumi, but she
is the daughter of Kaido. The kurozumi's were hunted because they were the family that killed a daimyo. Kaido also killed a daimyo. Oden nonetheless, a universally loved figure in Wano. Kaido's bloodline, (considering the precedent set with the Kurozumi's) would naturally be expected to face similar persecution.
The entire point is that Wano citizens were the ones that created the Orochi and Kanjuro problem out of their own spite.
Hence there needs to be a situation that proves that the citizens have changed. That they won't persecute innocent people again simply due to their bloodline. And that's why Yamato, being an ally, despite also being the daughter of Kaido, would be a logical route to explore that development. Evidence that they won't persecute Yamato and proof that she won't be the next Kurozumi Orochi, born out of the malice of Wano citizens. The best way to express that is by having Yamato live alongside them, showing their union with and acceptance of her, despite her bloodline.
The sexism angle- I don't think Oda treats that seriously. It's not a real plotpoint. It's a joke based on Kin being an old fashioned guy that's actually just trying to hide the fact he's a total pervert. If it is something to be answered, it will be done by Hiyori, Tama, Shinobu, Toko et al, all women from the mainland- who don't call themselves men due to obsessions with Oden.
Why? Yamato's the one in which her gender is actually a point of interest in her character. With the potential for her storyline to be to discover who she is and that she's not Oden, you don't think the fact that she's actually a woman that's deluding herself will have any significance?
And no, sexism isn't just a Kin emon gag. Quite obviously with Wano being inspired by feudal Japan in which women were generally seen as less important than men it makes sense that there are similar issues in Wano. Kin emon's not just randomly old fashioned. He's old fashioned because he comes from a country with dated customs. The idea of Wano being a nation lost in time is pretty clear. They're literally fighting dinosaurs, lol. The idea was even reintroduced with Kiku, a biological man, with the heart of a woman that thinks as a result of that she is to be protected and looked after by men, despite the fact she's fully capable of handling herself.