As Gamanit pointed out, it’s very unusual to have no clue who the birth mother is. The birth father is a little trickier, but in many cases, it’s a non-Jew. Even if the child was left on the doorstep, he can marry a convert. Likewise, a mamzer can marry a convert (though the children will be mamzerim).
Health, the term “real parents” is offensive to adoptive parents. There are plenty of reasons other than lack of money that birth mothers give up their children. Illness is one.
I think in many cases, children seek out their birth parents because their adoptive parents weren’t honest with them. From the time we brought them home, we told our kids that we had adopted them. They’ve never expressed much interest in finding their birth mother. I know of other adopted children who weren’t told until they were older, with bad consequences.