Does it leave open the possibility that God might still beget a daughter?
I think there are two issues at play here…first is the issue of God the Son as a person and then God the Son incarnate.
Check out the Baltimore Catechism online, that might be helpful for some clarification. I’ll just offer a few points.
First, God exists as one being in a Trinity: Father, Son, and Holy Ghost. Each are equal because each is fully the One true God eternal and infinite. Each is pure spirit because each is equally the one true God who is spiritual and non physical.
This means that Father, Son and Holy Ghost are not male or female.
Second, God the Son is not some separate entity who began to exist. God the Son exists from all eternity with God the Father. God the Son is equally God as God the Father. God the Son we say was “begotten not made” meaning he was more generated from the Father than willfully created. The Father had a choice in creating us, He naturally generated the Son by necessity not by choice. The Holy Ghost proceeds from the mutual love of Father and Son. Again, check out the Baltimore catechism, any addition (maybe 4 is the best) through a google search.
Third, God the Son (who is eternally begotten of the Father…genderless) became man through the incarnation. This means He took on a created human nature. God the Son is still eternally begotten of the Father however he took on a human nature through Mary as other humans do. God the Son exists as one person however He has two separate natures: The human nature (received from Mary) and the Divine Nature (which he posesses from eternity as does the Father and Holy Ghost)
Fourth, the human nature of God the Son is a male human nature because God chose to make it that way. He had a choice given that the human nature of Jesus was created in time. God could have taken on a woman’s nature however chose to not become a woman, he became a male human being. God the Son’s Divine nature (eternally begotten of the Father) is genderless though.
God couldn’t have begotten a daughter, as the begetting part is eternal and spiritual. God could have created a female human nature to become incarnate in however He did not