GovernmentShrinker-(irony)-a mode of speech conveying the opposite of what is meant.
The reality is that if people like this aren't prevented from having more children, the already huge government welfare and "child protection" bureaucracy grows ever more huge. It's not like they're going to become self-supporting, much less support their children. And of course, they won't bother to actually raise the children they have and don't quite manage to kill through violence and neglect, so when those children grow up, the government will have to provide yet more prison space.
Really, forcing them to get sterilized involves much smaller government than the alternatives.