time/place: New York City, 1993-2003
A cab driver goes to pick up a broken-down cab, and finds an abandoned newborn in the backseat. He takes her home with him, and his family looks after her for approx. 12 hours until social services (or is it Dept of Child Protection Services?) comes to take her (would they need to pass at least a cursory background check, for such a short time?). The child is placed in an orphanage, as a search for any parents, including a public appeal, is fruitless. The cab driver and his family keep in fairly frequent touch over the years ie birthday and christmas cards/presents, and she winds up good friends with the family's year-older son. However, they don't support her in any kind of financial or legal fashion.
When the girl is ten, various factors (irrelevant here) kick in, meaning she gets whisked off to boarding school, and signed into the custody of the Headmaster.
My major question is; the cab driver's family make several enquiries about adopting the girl over the first ten years, but are always advised that they would be turned down, so they never get to the formal adoption application. (though right before the girl is taken away to school, they put serious effort into it that they hadn't before) What kind of circumstances / reason would that require? They're a blue-collar Italian-american family, lower-income but not poverty-stricken in any way.
I did a search for NY adoption laws, and the website I found seems to think that the fact that there's no legal relinquishment signed by the birth parents won't be a problem, as long as they never make themselves known legally or to the girl. BUT would that be the case in the 1993-2003 time period?
EDIT: thank you so much for the help, everyone! I see I'm going to have to go alternate-universe with the orphanage idea (my character probably wouldn't get the kind of emotional detachment that's going to come into play later from foster care. As I understand it, foster families are usually meticulously screened, primarily so they won't damage the kids in this fashion.) My irrelevant-here-factors should take care of the rest.