There may be another source for Luke's account of the virgin birth. (You point out the obvious faults with Matthew's account.) It (and much more) seems to based on the life of an earlier prophet called Issa. It was claimed that he was born to a virgin named Mary.

I've written about it in this article.. It's very long, but the relevant material can be found about a third of the way in.

https://medium.com/@graham-pemberton/the-bible-came-from-arabia-669b0865e7c7