It's true that many parents are first-time parents, or first-time-having-a-bris parents (they have a girl or girls), but some people have incredible back stories that have brought them to this day.
For some parents it is an illness, or a spiritual journey; some babies had "issues" which were discovered by the sonogram or after birth, some babies are results of much fertilization medical intervention (a.k.a extra special miracles).
Some circumstances involve unexpected babies, either on account of age, or on account of medical history of parents. I've dealt with many IVF babies, a few surrogate babies, and a few adoption/conversions.
I've done hatafat dam brit on babies that were circumcised in a hospital, on adults who were circumcised in a hospital before the eighth day, and on adult converts.
It is always an honor to be brought into the inner circle, and a privilege to play a role in promoting the Covenant of God and Abraham, of God and the Jewish people.