This is a comparison with the manga:
In the scene where Shenjou talks to Iida, Shenjou doesn't say "Just like me". I think the manga handles it better because (1) it implies that Shenjou is conscious and shy enough to want to hide it, even if she's thinking about her own shortcomings and (2) it paints a better picture of Iida, who's smart enough and close enough to Shenjou to figure out that behind her criticism of Mina, she's actually thinking of her personal drawbacks.
Having it say it outright in the anime, I think, slightly diminishes the characters. It has less of an impact, in my personal opinion.
But registering the robot as a family means they get to do LESS work overall. I'm sure they'd love that. She says they still check "the most important part of the programming that relates to safety" (i.e the Three Laws). One would assume that there is the legal background so that is enough to waive any responsibilities. Asimov has written at least 2 stories where the robot scientist goes to court to explain how the positronic brain works.
Also, offering this type of service for free could be how there's only one robot company for Minas and nothing else. They offer the most so that they beat competition before it arises. See how services like Netflix and Uber started up, the first years they were extremely customer-friendly, until they established an almost-monopoly.
Mina's reaction is incredible and it's the first MAJOR indication that there's something like a conscience. Sure, the show did a few cute things like the "aah" and the heart omelet, but a sense of shame is a sense of sentient identity.
This always reminded me of Spice and Wolf, the transformation scene down at the sewers.