You're getting into some pretty deep philosophical territory there. When you say Jesus was the son of God, does that mean that he had 23 of God's chromosomes? Does God have 23 human chromosome pairs? If not, and if He chose which genes to give to Mary's baby, who says He didn't choose Joseph's?
Under Jewish law at the time -- and for that matter, under common law up to today -- a child born within a marriage is presumed to be the child of both spouses, absent proof to the contrary. So legally, Jesus was the son of Joseph. Ethically and emotionally, Joseph loved and raised Jesus as his own. What I'm suggesting is that this might also have been true biologically.
God is the author of life. I am pretty sure that He could create the genes required for God the Son to have a body of flesh.
Jesus is the Son of God as explained in various passages of the Bible. As per your Joseph statement, it’s just another way of saying Jesus is bioligcally Joseph’s son, which is wrong because the historical account is that Joseph had nothing to do with Christ’s conception. You also turn Joseph into a pious hypocrite because the record is clear he was going to divorce Mary, when he knew it was his child. Else you make god out to be a devious figure by taking Joseph’s genetic essence without him knowing it, and not telling Joseph about it, and allowing Joseph to think Mary was unfaithful and should divorce her. I think either way, your idea falls flat.
Till he was adopted into Mary's family...