Is it 1984 or Brave New World where the people are required to take birth control pills and all babies are incubated in "Malthusian bottles" and raised by the state? (Malthus was an early economist, if I remember correctly.)
The problem lies with the phrase "unfit parents". It's a slippery slope. Maybe we could all agree that someone who murders a child should be sterilized, but what about physical or verbal abuse? Is spanking physical abuse that should result in sterilization? Some people would think so, others would be horrified at the suggestion.
We would end up with government guidelines for "fit" parenting and anyone who chose a different path (conscientious objecters) would be subject to interference. I think of all the difficulties faced by homeschoolers in the last 25 years trying to get out of the system, esp. considering that it used to be the norm. Then multiply that by the total number of children in the US, not just a vocal minority. Or people in China who register to have children and are penalized for having too many.
Since we already require children to attend school (the only other group of people who is as restricted as school age children, is prisoners), perhaps parenting or even baby-sitting classes could be required? Add it to the health cirriculum, I suppose, right after they talk about how you get a baby in the first place.
I think it sounds okay in theory, but putting it into practice looks extremely problematic and fraught with opportunities for abuse and government control.