I don't know anyone whose family wasn't in some way odd
Whenever I hear of a newborn baby being found abandoned - and it happens everywhere - my hope is that the infant will be placed with an adoptive family as soon as possible.
I think it's cruel to try and hunt down the birth mother, when she has made it plain by her actions that she either doesn't want the child, can't cope with it, or just hopes to forget the fact that she gave birth.
Early life experience is hugely influential. A baby born in unfavourable circumstances needs a loving and stable family ASAP. Yes, it could be a loving and stable single person too: as Alan Shatter has written, what matters most in a child's life is attachment, and his novel Laura, though not likely to win any literary prizes, nevertheless describes a tug-of-love family situation he would have encountered as a lawyer. (And it's the adoptive family who emerge as the winners in this story.)