I honestly don't know how to respond because it sounds like you have some weird definition and requirements for story themes.
Themes don't drive the plot. They can, but themes are just central ideas/topics that are woven throughout the narrative. What you've dismissed as some minor thing in "a means to give a background, to justify certain behaviours" IS the meat of how themes work. Its in characters histories and lives, in how characters react to it, act on it, how it affects them and the lives around them. If you look at most of the central characters to BADIK's story, you see bio-family or self-made family affecting their actions and decisions, causing problems or solving them, and we see how the characters react to that and act on it. Thats how themes work.
Just look at the DIK house being trashed. Narratively its just "Jocks vs DIKs. DIKs need to find a way to scrounge up money to fix the place. MC helps them to do so." Now, looking it at it through the family theme, you have Rusty's problems with his dad. Being a shitty son finally takes its toll and it affects the self-made family he has, the DIKs. His real family abandons him to his fate. His 'brothers' are useless like he is, because thats the family he has built for himself. Rusty has had a rich, affluent life. He doesn't know what real life and real work is. And the DIKs having just been sucking on his teat, so neither do they. THATs how family has affected Rusty, his decisions, how its affected the DIKs and their decisions, and landed them in their current predicament.
And then you have the resolution of the house being trashed. MC, the outsider until now, the new brother. His family is just his dad, who is broke as fuck, the complete opposite to Rusty. And so MC's family has taught him to be resilient (unless he's being a whiny bitch over girls) and resourceful. And he begins to impart that onto the DIKs. Gets them jobs, show them how to be functioning members of the fraternity. Become men. And where Rusty's dad abandons him to his fate, Neil shows up to help the DIKs. Gets them cool shit.
Family informs why and how the characters got to this point, we see how it affects them in the situation, and even contrasts the relevant families at play in the situation (Rusty, MC, the DIKs). THAT is how themes work. And thats just looking at this situation through the lens of ONE theme.
I dunno, man. I have no idea what you expect themes to do in a story. But going by the regular definition and usage in literature, I really don't know how a person can look at BADIK and not see that family is the one of, if not THE, central themes to the game's narrative.
EDIT: Welp. This turned into a wall. So much for "i don't know how to respond", eh?
