I don’t remember the source, but it was a reliable Rav, at the time I met the person I mentioned. Of course rabbonim differ in their shitas, and circumstances may differ. I’m not sure how much a beis din would rely on current professionals.
In terms of a person who already keeps mitzvot and obviously knows what he/she is doing, it may be the case that Jewishness is assumed, but in the cases I’ve heard of the person was not raised Jewish and had to start from square one.
I lost track of this friend around twenty years ago, and at that time his wife, who was a religious Catholic, had shown no interest in converting, and his son was too young to deal with the question. He lived in a town were there were very few Orthodox Jews, and made the best adjustment he could, since he was absolutely not going to leave his wife or child. I hope it worked out, but I haven’t heard anything since long ago.