"Nice to contemplate" typically (always) becomes reality down the road; this becomes evident through studying all advances in healthcare systems, the acceptance of eugenics, euthanasia in certain countries, changes in legal liberties, etc.
Depending on which people were deemed worthy of deciding who can and cannot become a parent, I could be declared unfit, as could happen to every person in the US. That is the obvious problem with it, which everyone sees (well, for the most part; I've met several people that were honest proponents of the idea as a reality who failed to care about that). Who determines what is right? Who determines what is acceptable?
It is the same problem that plagues those who have a hazy definition of what constitutes human personhood. Something as amorphous as life cannot be quantitatively analyzed. I have no tolerance whatsoever for those who would condone killing infants that would grow up to be like my cousin or my uncle or my friend's sister. It is very hard for me to have a rational debate with people of that ilk because, though I have yet to ever yell at someone in my life and am certainly capable of remaining calm, every nerve in me wants to be the opposite.
What makes a fit parent cannot be quantitatively analyzed, either, because, again, it is very closely connected to life in a general sense. Abusive parents do not always show signs of such before they have children. Would it be prudent to ban all people who were physically abused by their parents themselves as children from having offspring because many times abuse is cyclical? A very good friend of mine was physically abused by her father as a child, to the point of breaking bones, and today she is a single mother of four children and is the best parent I have ever seen. In the small moments that she can see she's fully overcome her past in that sense, she credits her abuse with causing her to be more sensitive to her children's needs than most parents. She has never laid a hand on them, and they absolutely adore her.
Children are removed from unfit parents after they have proven themselves to be unfit. Before that, it seems to me that there is no way to call the shots based on "risk factors." Risk factors are just that: factors. They are not sentences.
How else would a person be deemed an unfit parent before actually having and raising children? Would it take three firings from a job to prove excessive irresponsibility? Four? Five? Should felons be prohibited from having children, just as they are stripped of their right as American citizens to vote? Should children not be allowed to be born in poverty? Are the IQs of the parents a factor?
I can't even entertain the thought of it because to me, it makes no sense at all. I mean, clearly I
have entertained the thought before in order to come to that conclusion, but beyond that... Researching the forced sterilization that this country has employed in past decades in addition to formulating my own thoughts based on my own experience has essentially made me unmovable when it comes to this. This is really all I'm going to say on the subject... It hits too close to home for me because several people I know who are all good parents have risk factors for being abusive themselves in their past so I don't enjoy debating it because every time I come across someone who supports it in earnest (which is maaaybe one or two people on this board; I wouldn't know who though, so don't take that as my having secret contempt for specific people or something

) I can't help but to think that s/he is incredibly ignorant and stupid and needs to take a look around at the people that he or she knows, and I don't like thinking of people that way, and really any criteria would be entirely subjective and not quantitative at all, so... Ehhh.
Not to mention that it's entirely impractical on a concrete, logistical level...
Edit:
I guess I should clarify that this post isn't directed at anybody. I just took the idea and ran with it.
Also for clarification purposes, I can also see the idea that
after having children and being deemed unfit as a result of abusing them, a person ought to be forcibly sterilized... I could understand that course of action. I do believe, though, that the sterilization should be reversible, because change is possible, and it's not something that I would actively support... I would still oppose it based on my own stances regarding personal liberty, but if opportunities were provided for change, my opposition would be more abstract than applicable to reality in the form of concerted protest, so in effect it wouldn't be opposition at all, and it would therefore not be worth mentioning. I am fairly certain, though, that there are already similar laws in place, at least in some locales... I recall reading several articles about women declared unfit mothers who not only had their current children but subsequent children as well taken from them. I am not sure where they were located, though.