Ms. Mizunokoji isn't a horrible mother to only Asuka. She's horrible towards Tobimaro too. Treating him like he's worthless.
I think the Mizunokoji family is matriarchal. The husband marries into the family. That'd explain why Asuka is getting trained like she's going to be the successor and the son is treated like he doesn't exist.
I think without Ataru's meddling Lum would've succeeded in helping the family.