In terms of political groups, fighting doesn't sort things out. It just hides it under a veneer of power. It can work for individuals, though.
A treasure hunt is rather different from a battle.
I get the impression there's not enough asking "Why?" in this manga.
Yes and No? This is really far from a true political group. For their part, they're just a gang. Part of why it's going to work, is that the Shadow Student Council doesn't respect the "Royal Faction". As they say in 95, they feel Marianne is idealistic and sheltered (therefore weak.). Aside from the plans of Alexei, they are the shadow council, because they think they're strong enough to go against the royal faction. They legitimately believe they have enough power and strength to work towards their goals. Now if the Royal faction straight out overpowers them, that means they never had the strength to do what they wanted.
Indiana Jones had a lot of battles for a treasure hunt. And these are supposed to be non-lethal battles. Not like they were ever going to go with Queensbury Rules for Fisticuffs.
I mean the why is the basis for her characterization? She's a kid with simplistic child-like beliefs, that's been Anime/Manga-Pilled. It's not like Char has any amount of adult life experience. All children extrapolate from what they "know" and try to apply it to their current situation. Is it still Chūnibyō, if you live in a literal world where you've got magic powers, and you've somehow become the center of the machinations of the world? Why wouldn't you apply Anime/Manga logic, when all you've encountered is an Anime/Manga World.