OMG lot of confusion and great debate. However if you ask me there are a few pointers
1) People do believe in miracles but only if it happened ages back, not if it happened now, for example if I say Abraham was asked by God to sacrifice his son, majority will believe, but if I say the same about my neighbor, they will consider him to hallucinating and a threat for the society.
Same thing would have happened back then as well. Divine birth theory might be making us bow to the Pandavas but back then Duryodhan took it as an opportunity to mock on it. Kunti decided to hide her pregnancy with a Kaneen child despite it being a divine child, because even she knew that people won't buy this theory. I therefore feel that Yagya kund se nikle bachhe wouldn't have been belief then
I personally feel that the proper adoption technique in those days was something to do with the fire. Maybe any oath on fire etc. which would seal the relation between the parents n the kids or anything of similar sort. Such children would have been called the kids of Yagya Kund, since they became the kids only after the oath of Yagya Kund. I am into this theory because there have been many kids mentioned as arising from Yagya Kund the most important excluding the Draupadi Dhristdhyumn twin being the Ila who originally was the Yagya Vedi se nikli daughter of Manu but later became one of his wives
Rest about biological parents what you are saying seems pretty valid. But then as CS mentioned who was dad on whose lap Draupadi spent her time in childhood. Secondly Draupadi herself was very learned girl, how did she get that knowledge if she was adopted as adult and her parents were not so rich