Yuri’s idea of “helping the little people” works because he IS the little people. Everything he does is done out of necessity, because Abyssians are very much a marginalized group. Doing bad stuff, in Yuri’s case, is a means of survival because someone like him has little other choice. Edie, however, does Bad Tings™️ in order to get power and enforce her messiah complex on all of Fódlan, DESPITE having other options available to her.
I don't know if I would word it quite like that but yeah, Edie thinks she's the only one who can change things and that her way is the only way. It's hard to tell how much of that is her being naive and sheltered, the Agarthans manipulating her and how much is her genuine belief that there are no good people
That's probably also why Dimitri gets such a warm welcome in Fhirdiad despite the things he's done. Because he was basically downgraded to living on the streets for a while so he actually knows what's what and they know he knows this.
And Claude has had to fight for a place at the dinner table his whole life too, despite being born a goddamn prince. I mean, he's a master poison brewer at age 17, he knows how shitty life can be.
As for Yuri, he has been bouncing back and forth from little to big for almost two decades, and he's used that knowledge and influence to help those who actually need it. He can flawlessly inset himself into any company he wishes.
Edie doesn't have that, she doesn't really know the difference between herself and the little people. She probably thinks being on the opposite end of the "food chain" is what happened to her and her siblings so she believes she knows better than she does
And because she can't imagine that Dimitri or Claude would understand it better than she does, which they do cuz they've both had their share of bad both before and during the game, she doesn't think they'd be able to enact the changes that are needed.
She lacks the insight needed to actually do what's right, but she doesn't know it. The other house leaders, and especially Yuri, do but unfortunately they're not the ones who decide where the story goes.