Why listen to her? Just take the skill and find out yourself, if he wont take it, it will bite them in the ass later, that girl is bad news.
Because she's in it for herself, and that's a very good thing.
Lemme elaborate.
Think about what the best route is for a very selfish person in a setting like this. It'd be to gain absolute authority over a nation that is strong in all areas - politically, militarily, economically, and sociologically.
But that's not likely to happen for her, no matter how much manipulation she pulls. Sure, she could get herself married off to a suitably powerful figure of authority by being their personal soothsayer and trophy wife... but I suspect she's not gonna spread her legs for that reason. She'd still be something of an underdog, being beholden to her husbando and all.
The next best thing is to be an indispensable person among the leadership in such a country.
It's obvious that since she seems to want to be on Yuto's side, the Valencia Empire is on its way out. (Excuse me, here, but this tangent is needed for context.)
This is typical of Empires that prosecute war after war like Valencia seems to be doing. Even if you're successful in your conquests, you spread yourself too thin. Especially if you're bent on genocide as these human supremacists are, because afterwards you no longer have anyone to tend to the land. It's all empty and useless to you, and you're now trying to hold that territory from your enemies. With soldiers that you have to feed and equip from your nation's own resources, and not that of the territory you just took.
Oh, sure, there's spoils alright. But those are only good for so long. They give you a momentary boost, and that's it. Without a population to continue producing resources, you're back to square one pretty damn quickly. And there's also the fact that they need to replenish their ranks of foot soldiers. Because even if they've been winning, they've still been taking casualties.
Thus, they've been summoning Heroes to bolster their ranks and their capabilities. But that doesn't always work. Because if you get someone who has even a bit of moral fiber and the courage to stand up against you, then you're in trouble.
Granted, in the case of Yuto, he had to be abandoned because he was such a conformist that he'd have never have left otherwise. And even then, the Demon Princess had to guide him along. But still, it happened. And because the Empire was enabling the more spiteful and avaricious personalities, it's caused the group of summoned Heroes to fracture in the worst way possible.
The Empire is going to fail, and mainly because they prefer toxic personalities that will destabilize their foundations.
Now, getting back to this girl...
She's seen the writing on the wall, so she's not going to stick around on a sinking ship. Granted, she could do her part to make sure the Empire is victorious, but as I said, it's not a matter of if they'll fail, but when. And the kind of reform that would prevent this won't be able to come quickly enough, mainly because of the people in charge.
So this leaves Yuto and the Demon Princess. It will be in this girl's best interests to side with their group and do what she can to ensure their victory over the Empire. The only other option is to be treated as a tool for the Empire and/or some nobleman's breeding sow.
She's in it for herself, for sure. But that just means she's gonna support the side she thinks will win. It's very unlikely that she'll betray Yuto.
I just hope that she convinces the other girls to not go back. Hearing that bit about Miyamoto's "Harem" does not bode well for them. That dude is an obvious narcissist. And I mean that in the actual, clinical definition of the word. They're the absolute worst kind of psychopath to deal with, because not only is everything they do for their own selfish gain and aggrandizement ("Worship me, for I am the Hero among Heroes!"), but they'll actively seek revenge against anyone who pricks their pricelessly precious ego.
Minase will not last long dealing with someone like that... that is, if Miyamoto doesn't break her mind and soul, first.