Purpose of the Virgin Birth

I congratulate you on having the courage to drill down on the Virgin Birth.

However, you missed a key passage in your analysis:

Is 7:14:
"Therefore the Lord Himself will give you a sign: Behold, a virgin will be with child and bear a son, and she will call His name Immanuel."

The fulfillment of this and further explanation is here:

Matt 1:18-23
18Now the birth of Jesus Christ was as follows: when His mother Mary had been betrothed to Joseph, before they came together she was found to be with child by the Holy Spirit.

19And Joseph her husband, being a righteous man and not wanting to disgrace her, planned to send her away secretly.

20But when he had considered this, behold, an angel of the Lord appeared to him in a dream, saying, “Joseph, son of David, do not be afraid to take Mary as your wife; for the Child who has been conceived in her is of the Holy Spirit.

21“She will bear a Son; and you shall call His name Jesus, for He will save His people from their sins.”

22Now all this took place to fulfill what was spoken by the Lord through the prophet:

23“BEHOLD, THE VIRGIN SHALL BE WITH CHILD AND SHALL BEAR A SON, AND THEY SHALL CALL HIS NAME IMMANUEL,” which translated means, “GOD WITH US.”

Notice, Mary conceived a child with the Holy Spirt being the Father and the name of the Child is עמנו אל (Immanuel). "God with Us"

So two purposes for the Virgin Birth are stated here:

1. To accomplish the incarnation of God the Son as the God-Man, Jesus Christ. God the Holy Spirt supplied the divine Y Chromosome and Mary Supplied the human X Chromosome.

2. To save the people from their sins. In order to be the Sin Bearer, Jesus had to be "without blemish and without spot" (1 Pet 1:19) meaning he could not have a sin nature. The Virgin Birth accomplished this as the sin nature is passed down through the human male in copulation:

For as in Adam all die, so also in Christ all will be made alive. 1 Cor 15:22